Is are there another database server software for PHP other than MySQL?
Does it run on Linux?
Does it work with Apache?
(No MySQL, can't get it working on ubuntu.)
Yes. There are hundreds, perhaps thousands of database servers available to you.
To be frank, if you haven't been successful in getting MySQL working on Ubuntu, you're either:
MySQL should be trivial to get running on Ubuntu. Quite honestly, ~30 seconds to install (and that's if you have a very slow internet connection), and perhaps 30 minutes to go through the steps of creating your first database and creating a user that has privileges on that db.
Other popular database server options are:
None of these (perhaps with the exception of SQLite, which isn't widely-supported) will be any easier to get running than MySQL.