"Alone" is grammatically okay since it can serve as an adverb, but as others have said the meaning could be ambiguous. "On my own" would sound more natural.
The Present Perfect tense (or the Present Perfect Continuous tense) is needed in that construction because "since 2022" is a time period that we're still in. A lot of adverbs and adverb phrases trigger the Present Perfect for that reason: recently, lately, in the last few days, etc.
I would recommend:
I'm learning English on my own. (no time frame is specified)
vs.
I've been learning English on my own since 2022. ("since 2022" is a time period that we're still in)