A server should run just fine on that system - I ran my web server on a low end P4 with less RAM than that and it worked fine. It won't support thousands of concurrent users, but for testing it should be perfect.

I would think that you could use SFTP to replicate your copy of the site to the live server. It's not the ideal solution, but it should work. I do wonder if you'll have permission on a shared host (which is what it sounds like you're using) to do uploading through the web at all, but there might be some way. The only thing that I tend to have problems with when moving sites between servers is when pages have references to absolute addresses, in which case changing the server address breaks things. How big a problem that will be depends on the site.