I agree. Another way you could say it is "I'm great at fixing things." Here, you can use "great" as an adjective, although you are describing yourself rather than the work directly. This is a bit less formal than the examples above.

I would probably use "I can fix things pretty well" It would have to be an adverb, since you are describing the verb, but we don't use "greatly" very often. An example I can think of using greatly would be, "I would greatly appreciate it if you could help me with this". It sounds a bit more formal.