Never had a threadless fork before, do I need a star nut inside the fork? How much does a star nut cost?
How much is it to have a star nut installed? Only if it requires a special tool or is quite difficult

Yes you do need a star nut, should only be a couple of bucks. There's fancy tools available in LBS to do it, its very quick and therefore probably cheap, but some care with a hammer and drift will get it installed too.

LBS should install it for a few bucks. They can do the crown race at the same time. Last time I had a crown race put on I was charged $2.