There are 2 kinds of git repository, bare and non-bare. A non bare repository is any repository which has a 'working copy' i.e. some part of the repository currently checked out.
You can push into a non-bare repository, but it won't update the checked out working copy even if the checked out branch is the same as the branch you pushed. This is because the checked out copy might have changes that aren't committed, and git won't ever destroy changes without you explicitly asking (usually such commands have a --hard argument)
Read Why won't I see changes in the remote repo after "git push"? and How would I use "git push" to sync out of a firewalled host? for a full description of the problem and a potential solution. Word of warning, if you've pushed into a remote repo, any non-committed changes in that remote-repo will have to be discarded.
Generally it sounds like the approach you want isn't really adopted by gitters, because it doesn't really match the distributed repository mentality. It's your own responsibility to make sure your copy of the repo is up to date.