My Mac mini is currently on version OS X 10.4.11, how can i upgrade it to OS X Mountain Lion?

Open About this Mac from  What does it say for Processor?

First check that your Mac Mini can run Snow Leopard:

Requirements for OS X 10.6 'Snow Leopard'

http://support.apple.com/kb/SP575

Whilst Apple have withdrawn Snow Leopard from their stores, you can still get it from Apple by calling 1-800-MY-APPLE (if you are in the USA) and they will supply the SL DVD for $20 for a single user, or $30 for a family pack that covers up to 5 Macs. You can also purchase the code to use to download Lion from the same number.

If you are outside the US call your national Apple Helpline:

http://support.apple.com/kb/HE57

If you're in the UK, use this number: 0871 508 4400

When you have installed it, run Software Update to download and install the latest updates for Snow Leopard.

Sorry, Lepard 10.5.8 is as far as any PPC Mac, like your G4, can go.

You can get a black retail install disk for Leopard on eBay or Amazon - at a price.