Rather than printing the photo on paper the full size of the frame, you'll probably prefer the results, along with the significantly lower cost of the print, if you cut a matte for the border and only print the photo large enough to fill the hole in the center of the matte. The cost of the matte will be minor compared to the savings gained by not paying for a larger print that is mostly unprinted white border.
Short of matting, you could place a piece of A1 sized paperboard behind the print of whatever size you desire. You'd use two sided gaffer tape to hold the print to the paperboard and then place the combination in the frame. Use a thick enough pad (usually corrugated cardboard) behind the paperboard and the frame back to hold everything tight against the glass. Again, the cost of a print only as large as the size you wish the photo to be in the center will be noticeably less than the cost of a print on A1 size photo paper.
(Please forgive me in the following section for thinking in outdated Imperial units, but that's how we do photo related things here in the U.S. It's also apparently how most photographers and photo finishers selling enlargements still think in the UK, which is where the website selling the frame you've linked is based.)
How would I calculate the width of the border to add? Is there a standard?
There's no real standard for the ratio of matte sizes to print size or frame size.
You'd calculate the size you want the borders to be by subtracting the size you want the area that has the actual photo in it from the total size of the frame. You'd do this independently for both width and height. Divide each result by two and that would give you the thickness of the border on each side of the photo printed in the middle. If you're matting, increase the border size by about 1/2 inch (12 mm) to allow the matte to overlap the photo by about 1/4 inch (6 mm) on each edge.
What often happens is the print size plus the size of the desired borders determines the needed frame size. A matte is then created to allow an standard sized frame to hold a standard sized print that allows the width of the borders on the matte to be roughly equal.
For example, an 18 x 12 inch print with a matte having 3 inch borders¹ will need a 24 x 18 inch frame. Both the 24 x 18 frame and the 18 x 12 print are easy to find sizes without the need to get a custom sized frame or print done. So are pre-cut 34 x 18 inch mattes with 18 x 12 inch holes in them.¹
¹ The holes in mattes actually allow an extra 1/4 inch or so of overlap of the matte over each edge of the photo so the photo is secured behind the matte and doesn't fall through the hole. So a pre-cut 24 x 18 inch matte made to hold an 18 x 12 inch photo actually has borders that are 3 1/4 inches per side and leaves a 17 1/2 x 11 1/2 hole in the middle.
Sometimes an already selected frame is used to hold an already printed photo and the dimensions of the matte are determined by the differences of the frame and print sizes. In general, mattes shouldn't be narrower than about 2-2.5 inches (50-64mm) on any side of the photo, regardless of frame and print size. They can, however, be much larger than that. At that point it's all up to personal taste (or the existing dimensions of the frame and print you already have).
Viewing distance is usually calculated to be roughly equal to the diagonal of the photo itself (not including any border or matte). The traditional "standard viewing conditions" assumes an 8 x 10 inch (20 x 25 cm) print viewed from a distance of 10-12 inches (25-30cm) by a person with 20/20 vision. An 8 x 10 inch print has a diagonal of 12.8 inches (32.5cm).
There's a general standard that says if a print is intended to be viewed from a distance of around 10-12 inches (25-30mm) then it needs to be printed at a minimum of 300 ppi (pixels per inch - which translates to about 118 pixels per centimeter). Your 5938 x 3959 pixel photo at 300 ppi would be 19.8 x 13.2 inches (502 x 335 mm). An A1 size frame is 841 x 594 mm (33.1 x 23.4 inches) and a matted/bordered print in a frame that size probably will not be viewed from that closely, so you have some leeway with regard to how large you can print the actual area containing the photo. If viewed from twice as far away, 150 ppi (1/2 that of 300 ppi) would be the same angular resolution to the viewer.
At this point we must consider that your photo has an aspect ratio of 3:2 (1.5:1) while your intended frame has an aspect ratio of roughly 7:5 (1.4142:1 or √2:1). This means the borders along the long sides of the print will need to be wider than the borders along the short sides of the print.
For example, if you printed your photo at 841 x 560.7 mm (you wouldn't), you'd have no border on the short sides because your photo's long dimension would be the same as the long dimension of the frame. Your photo's short dimension would be 33.33mm narrower than the short dimension of the frame. Dividing by 2 and allowing a bit of overlap, you'd need a matte/border 17.3 mm wide on each side of the photo. Of course that would not be very pleasing visually. But we now know that two sides of the border will need to be at least 17 mm (≈ 2/3 inch) wider than the other two sides.
Let's say you were to print your photo at 24 x 16 inches (609.6 x 406.4 mm). That's a fairly standard print size here in the U.S. (and I see it mentioned on several websites based in the UK as well). You'd have a photo with a diagonal of 28.84 inches (732.5 mmm) printed at about 250 ppi (99 pix/mm). 250 ppi would yield the same angular resolution to a viewer at a distance of about 14 inches (36 cm) as would a "standard" 300 dpi print viewed at the "standard" distance of 12 inches (30.5 cm) by the same viewer. We can see that 250 ppi is more than enough for a 24 x 16 inch print with an assumed viewing distance of about 25 inches (63.5 cm).
Allowing for about 6 mm (1/4 inch) overlap between the aperture of the matte and the edges of the print - so the print doesn't fall through the hole in the matte - you'd need an A1 (841 x 594 mm) size matte with a 23.5 x 15.5 inch (597 x 394 mm) hole in the middle. That would leave borders of 4 13/16 inches (122 mm) and 3 15/16 inches (100 mm) on the short and long sides, respectively, of your print.
If you wanted more border than that, you could get the photo printed at 18 x 12 inches (457.2 x 304.8 mm). That size is available here in the U.S, I'm not sure about the UK. That would leave you with borders of 7 13/16 inches (198 mm) and 5 15/16 inches (150.5 mm), respectively, along the long and short sides of the photo.
I've placed 18 x 12 inch prints in 24 x 18 inch frames with mattes that have 3 inch borders and they look nice. I once created a 24 x 18 inch matte to crop an 18 x 12 print so that only 18 x 11.1 (the golden ratio) was visible. The long sides had 3.45 inch borders while the short sides had 3 inch borders, a 1.15:1 ratio, and it looked fine. Placing a 24 x 16 inch print in an A1 frame would give you a 1.22:1 ratio between the width of the borders. Placing an 18 x 12 inch print in an A1 frame would give you a 1.32:1 ratio between the widths of the borders, and might be pushing it just a bit.