일본어 공부를 한달 넘게 시작했는데 점점 좋아지고 있다. 1. 일본어 공부를 한 달도 더 전에 시작했는데 점점 좋아지고 있다 = I started my Japanese study more than a month ago, and it is getting better little by little. 2. 시작한 지 한 달도 넘은 일본어 공부가 점점 좋아지고 있다 = The Japanese I started more than a month ago is getting better with time. The original is a little unclear in a couple places. 1) 한 달 넘게 (more than a month) may be used for continuous activities, as in 한달 넘게 (X를) 하다 = have been doing (X) for more than a month. It can't be used directly with 시작하다 since 시작하다 is not a continuous activity with a duration. 2) 좋아지고 있다 (getting better) sounds slightly vague. If your study is progressing well, 잘되고 있다 may be better, You could also say 점점 (더) 재미있어지고 있다 (is getting more interesting with time).