It is a Modern Orthodox school. It’s part of Yeshiva University (on their campus in Washington Heights), which can be a great bonus (for the record, I’m a current YU student). There is a dorm as well as buses to local communities. It is definitely on the “modern” side (a strong part of its hashkafa) but it does have a diverse mix of guys. It will probably make a big difference which shiur you end up in.
If you’re looking for something even somewhat yeshivish, it may not be the right place for you. Some of the teachers are yeshivish, but the school’s official direction is not. You will not be in the yeshivish community (for better or worse)- though of course you can probably do whatever you want after graduation.
Most people I know who went there had a good experience- I would strongly advise contacting their office and maybe going in for a tour- that way you can see for yourself.
If you are looking for something more right-wing, then it’s probably not right for you. What exactly do you have in mind as far as how right-wing? Why do you want something more right-wing? What attracted you to MTA? I would advise looking into Shaar HaTorah in Queens or Ohavei Torah in Riverdale based on different clues you’ve given here, but I could be totally off base.