It's very simple. The full phrase of "мне не привыкать" = literally, "[I faced such a stuff many times yet, so] I don't need to get used to it". or, "I don't need to be getting used to it [because I already got used to it]". Phrase parts in the [ ] is meant, but not prononced. I.e. the meaning of the saying you're asking about is "It isn't new to me" or "I have faced it yet", or "I did such things many times yet".

"Но мне не привыкать" literally means "but I don't need (time) to get used to it," so basically, you're right, the meaning is "I'm used to it, I'm familiar with it." Note that we often use this phrase in a bad situation. For example, "Мне пришлось самому ставить запаску, но мне не привыкать. [=I had to change a flat tire myself but I'm used to doing that.]"

It is already the way it is Done or experienced something before (similar or harsher) - been there, done that It no longer seems to be out of order (though it may be unpleasant or not suitable and it is likely to upset or offend people) It doesn't surprise anymore Конечно же можно сказать: мне не привыкать, тебе не привыкать, нам/ему/ей/им...I think you know it. P.S. - Oh, there is no hot water again. It's been three days! Well... why doesn't it surprise me any more?) Enjoy your day!

This phrase doesn't always has to have a bad connotation. We also can use it in a joke, so there are 2 most common situations: 1 - Something that kind of upsets you happened and you're not really happy about it: A guy stole a book from the store where he'd already stolen it 4-5 times and now got caught again. His mom's friend tells her that she's sorry about the whole situation and his mom replies:"Что ж, мне не привыкать" ( "Well, I don't need to get used to that (because I'm Already Used to situations like that)) 2 - Not really bad situations when you can't do anything about it but laugh: Colin broke his arm or something else for the hundredth time and before that happened he had many other cases when he'd broken something. He sure has great friends who came to visit him:"Jeez! Look at you! But dude, honestly, тебе не привыкать" ("You don't need to get used to it" because he broke so many parts of his body that he should already be used to it)

"Мне не привыкать" implies "Мне не (нужно) привыкать (к этому)" It's something like "I don't need to get used to it because I've already experienced it and got used to it" :)