Gears on a singlespeed frame
It could work, but you'll end up with unsightly cable ties or something to hold the housing to the frame.  Plus the shifting performance won't be great given all the slop that can accrue in a long unsupported housing.

If you're fixed on this frame, how about getting an opinion from a local bike shop or frame builder, as to whether they can braze on some... "braze-ons"   That was how the extras were fastened on in the early days.  Of course all this comes at a direct cost to your wallet.

Price this against an internally-geared hub like the nexus and make a decision based on price.

Or learn to ride a single speed and save.