Since the question edit, I'm tempted to link across to my previous answer on the 'medieval look' which does cover a lot of the same ground in a reasonably simplistic way.
How can I create this 'medieval look' using an entry-level camera like the Nikon D3300?
Matt Grum's answer on How can I get dramatic shallow DOF with a kit lens? covers it very well too.
Original answer below...
I'm not, tbh, seeing a particularly shallow depth of field, mainly it's all fairly evenly out of focus.
On a modern lens you get a shallower DoF by opening the aperture, or using a longer lens from further away, or both. Alternatively, similar framing with the same lens on a larger format sensor will shorten the DoF.
Rather than strictly being concerned with the DoF, I found I could do a crude mockup to make your new photo look more like the old one just by pushing the contrast much harder, blurring & adding a dark vignette. I also removed what little colour remained in it - it was very slightly 'sepia' before.
1 minute in Photoshop gives...
Better results could be gained from a larger original & a bit more care & attention than my quick attempt. My vignette is definitely too crude.
Force-matching the light source of the original would take considerably more effort, lot of dodge & burn, by hand.
I had a bit of a go at 'washing out' the background by evening out the tones a bit like the old photo, greyer to the left, blacker to the right - the original gives me the feeling the photographer held up a [pyrotechnic] flash high & wide in his left hand.
I'm not sure it was massively successful, but I thought I may as well post it rather than just bin it... It could possibly benefit from a tighter crop.