Unknown 50s/60s track bicycle

Regarding the handlebars:

There are things you can do. First clean it with brake cleaner. Easiest way to remove the dirt and the old glue or possibly even shellac. Then use Scotch Brite to get rid of all the scratches. Do not use sandpaper or anything.
If this was a newer type of Bike I would suggest polishing this after the scotch brite but this this, I don't know. Might look better just to clean it and use cork or cloth tape at the bottom of the drops.

Thanks, I will try this!

Regarding tape, I ordered this: dirty white/crème/brown/whatever