Can I host WP-admin locally for a remote blog?

“MySQL server has gone away for…”

This should never happen as result of admin operation on vanilla wordpress. One post publishing should not include enough DB writes to make the DB go away. Have you tried with all plugins inactive?

Hey, is there a way I can host all the WP-admin functions locally, and simply push changes to the remote server?

Theoretically yes, you can use rsync or any other file sync software to sync your files (just the uploads directory or maybe even the full WP install) and some DB sync tool (I’m not an expert so will not try to recommend) to sync the DB

Two things to remember to do if you will try it

  1. Configure your hosts file to have your sites domain to point to your PC
  2. Depending on the DB sync tool you either will have to ask your host to let you access the DB server remotely an idea which your host might not like.

IMO the effort you will waste on investigating and implementing such a solution will better go into finding a better host.