Yes, you can clean out the fixed cup with it in the fame; just reach in with a rag from the other side.

Good luck finding a new seatpost--it's getting nearly impossible to find anything below 25.0mm except 22.2mm (7/8") and 23.8mm (15/16" and this one won't be any better than your current one). Unless you're willing to spend $70 on a modern 24.0mm post (http://www.yellowjersey.org/frogstem.html), your only bet is a 22.2mm BMX post with a custom (or homemade) shim. If your current post works, use it.