Am I using "apt" correctly?

I was iffy, although it sounded right. I didn't google the definition, but the phrase apt problem solver.

Weird Googlism here: If you google that phrase, without quotes, you get a bazillion hits related to problems with apt, the Linux packaging tool.

If you Google it with quotes: "apt problem solver", you get 640 hits, but only the first few are about Linux. The rest are using the the phrase exactly as you are.

So, as _Object already said, yes it is.

No, I'm afraid it is not a correct usage.

One meaning of apt is "appropriate", and the sentence "I am an enthusiastic and appropriate problem solver" does not make sense.

The second meaning of apt is "likely to undertake something". It does not imply "likely to do something successfully"

You could, for example, say "I am an apt [appropriate] person to engage for solving puzzles" or "I am apt [have a tendency] to go off and solve puzzles" or "I am adept at solving puzzles". The misuse of apt instead of adept is often called a malapropism. Mrs. Malaprop was a character in a play who habitually used not quite the right word. Her name comes from the French phrase "mal à propos", which means inappropriate.

Although the usage may be correct in the sense of "a quick learner", "adept problem solver" is likely your intended meaning.

Totally inapt.

Yes it is.