How to tighten sink faucet?
the nut around that brass threaded pipe is how to tighten it? Is tightening that the right way to go?


YES and YES


  do you think I can get it tight enough just with that? Or should I expect to have to do something extra (plumber's putty?)


YES and NO - you should go ahead and try tightening it up with the nut, it should work. Those types of faucets normally have a gasket under the upper portion and no plumbers putty is required.

NOTE- it will likely be difficult to tighten that nut, most of those types of faucets come with a manufacturer-supplied special tool to reach up and tighten the nut with. If you don't have that tool then you will need to improvise.