4 months is a decent amount of time to prepare for BITSAT. However, you’ll have to make a proper study routine and stick to it.
I also had just 4 months at hand when I seriously sat down to study for BITSAT. By seriously I mean that by that time I had a fair enough grasp on basic concepts. I studied for long hours and covered the NCERT syllabus thoroughly. I read and reread each chapter, till it stuck in my mind, and regularly practiced mock tests. My mock test scores kept on improving with each practice session.
I had thought I wouldn’t make it into BITS, but I ended up getting BE Mechanical at BITS Hyderabad. So I would recommend you study with diligence and solve mock tests on a regular basis. If not 350+, you’ll definitely score decent enough marks to make it into BITS.