Present simple is incorrect because 'learn' cannot be always true. You need to say when 'learn' happens. I learned or I have learned - in the past I am learning - at the moment I will learn or I am going to learn - in the future Also, using 'punch needle' as a verb sounds odd to me. It is usually a noun (like embroidery) so needs 'do' before it. 'Punch needling' would be better though. 'I (learn in the appropriate tense) how to do punch needling on my own' would sound more natural. Instead of 'learn on my own', you could also use 'teach myself' - 'I (teach in the appropriate tense) myself how to do punch needling.

It’s wrong and unnatural because it is in the wrong tense. It’s also not understandable to many English speakers (like me) because ‘punching needle’ isn’t part of our vocabulary. Correct is: I’m learning how to punch needle on my own.

Yes it does sound unnatural. In fact I'm not even sure what you are talking about. What is the context? Can you say it another way?

"Punching needles" is not an English phrase, my best guess is that you are talking about "sewing" here - the act of joining two pieces of fabric together with a needle and thread. If that is the correct context, then your sentence should be "I am learning how to sew on my own". The phrase is ambiguous, however, and could mean "I am teaching myself how to sew", i.e. I don't have someone teaching me how to sew, OR it could mean "I am learning to sew solo" i.e currently you can only sew if you've got someone helping you, perhaps to thread the needle for you, and you are learning to do it without someone else's help.