The magic phrase you need is "gear selector plate". Search this forum for that phrase and have a read. This is the likely source of your troubles, it is a very cheap part. If you just want to replace the bushes, even cheaper! Like, less then £5 cheap!
This part sites under the battery in the engine bay, and is easily less then an hours effort for doing it at home with very simple tools. At a garage, even quicker!
If it is the part under the actual gear stick that is worn, it involves more work as the exhaust will have to come off to get at it from underneath, so slightly pricier! Nearly always the other bit though, as the forum search will reveal.
Not a £300+ job, although a garage not familiar with Alfa's may not know this.