If its friction (and it almost definitely is) you can use any cassette or freewheel so no you don't have to use a screw on. 10 gears might be a bit fiddly on friction though.
You'll probably need to respace the frame for a cassette wheel though.
Also that frame has no hanger so you'll need a rear mech with attached hanger. SS would be easier and as its a pub bike you're unlikely to need a range of gears.
If its friction (and it almost definitely is) you can use any cassette or freewheel so no you don't have to use a screw on. 10 gears might be a bit fiddly on friction though.
You'll probably need to respace the frame for a cassette wheel though.
Also that frame has no hanger so you'll need a rear mech with attached hanger. SS would be easier and as its a pub bike you're unlikely to need a range of gears.