1. This year may I wish you finally learn how to use your smartphone properly.
2. This year may I wish you finally learn how to use your smartphone properly?
[Do we have to use a period or a question mark?]
a period I would think.
Yes, a period. It's not a question, it's a statement. You wish this will happen. It would be a question if you ASKED, "This year, can you learn to use your smartphone properly?"
To determine whether to use a period or a question mark at the end of a sentence, you need to consider the sentence's function.
1. "This year may I wish you finally learn how to use your smartphone properly."
This sentence is a statement or a wish, expressing the speaker's desire for the recipient to learn how to use their smartphone properly. Therefore, it should end with a period.
2. "This year may I wish you finally learn how to use your smartphone properly?"
This sentence is a question, asking for confirmation or seeking information. Therefore, it should end with a question mark.