Yes, I use both.
'Not my cup of tea' is more old-fashioned, but it's by no means outdated, and is definitely still in current use. It has to do with taste. For example, if were asked what I thought of someone's newly decorated lime green and sugar pink living-room, I might say 'It's not my cup of tea', meaning it's not a style that appeals to me.
'Not my thing' is a more modern expression, and it has a wider use. It can refer to taste, or to an interest in general. A 'thing' can be anything that is associated with a person - something that you do or enjoy. For example, you can say 'Animal rights is John's thing', meaning it's a hobby, a passion, or something that is important to him and he identifies with. 'Not my thing' is the opposite of this.