It might be more natural to say "I don't want to get any older" or "I don't want to be a year older". Hope this helps, and happy New Year!

come and gone with every year.

Thank you so much for letting me know the right ones! It helped me a lot with my confusion. I really appreciate it!

You could say also say that you don't want to age or you don't want to grow up both have slightly different meanings. The first refers more to physical age, the second more to outlook on life or mental age ..you want to remain as a child...