Is there a rule-of-thumb or some sort of strats to conider when doing the minigame? I know its not recommended for beginners but i cant find anything on it and when ive watched runs in the past, runners just blast through the minigame.

There are some patterns but it mostly comes down to having practiced it a lot. Early on don't worry about completing the puzzle fast. Just focus on completion and speed will come with time.

Tutorial videos on how to solve a 15 puzzle can help. Some runners get a puzzle app on mobile in order to practice on the go.

Since you mentioned PSP specifically, for the regular Any% you can actually choose to skip it entirely since with the overpowered gear you can deal with the encounters more quickly and the reason runners attempt the puzzle is mostly for a 1/6 item drop in 2nd place.