Your questions are very general. However, to give you some general responses, you can run MySQL on any version of Linux, on MS Windows, Mac OS X, etc. If your client's server is going to have Linux on it, you don't have to have Linux on your computer to interface with it. You can use something like secure shell (ssh) to log in and then use the mysql client (notice that mysql is all in lower case letters) to access the MySQL server.
As for MySQL's abilities, it can easily replace Access. There are various methods by which you can convert their Access data. You can use the MySQL Migration Toolkit to do this:
http://www.mysql.com/products/migration-toolkit/
To manage the data in MySQL, since you're used to a GUI interface like Access, you might check out the MySQL Query Browser. It will run under Windows:
http://www.mysql.com/products/query-browser/ You might also want to use the MySQL Administrator to administering the data, rather than using and querying the data:
http://www.mysql.com/products/administrator/
If you want to deal with MySQL in a more direct manner and want to learn it, you probably should get a good book for learning MySQL. I'd recommend "MySQL Tutorial" (MySQL Press) or "Beginning MySQL Database Design" (Apress). The first is easier, the second one is better.
Russell Dyer
Author of "MySQL in a Nutshell" (O'Reilly 2005).