Okay, so I am not a fan of going to hospitals or doctors appointments in general. Mainly because I am not adult enough to make my own appointments. Unfortunately I was injured not too long ago when I messed up my meniscus. Luckily, going to the doctors/hospital was very easy and cheap compared to the US. One thing that helped is that a friend gave me a ride and handled all the insurance stuff. It was just more efficient and they were trying to be nice/hospitable. In the end, I got everything I needed relatively quickly. This is after I went to the quack doctor in our neighborhood who wanted to give me a Benadryl and call it a day.
In general, the hospitals and doctors in Amman are good. This is from friends' combined experiences (I've only gone for my knee). There are specialty hospitals like private hospitals and general ones. I went to the one in the Gardens but they didn't take my type of insurance. However, I really liked my doctor so I decided to pay for his consultations out of pocket (it was cheap). However, for the expensive stuff I went to a hospital further away that took my insurance. I forgot the name. Anyway, both were good. The main point, any routine healthcare that you need you will be able to find it easily in Amman.
If you don't speak much Arabic don't worry because its not a problem. One thing worth mentioning is that you will need to take your passport with you. Also, if you're on birth control and coming to study in Jordan for a short time, it may just be easier to bring enough with you for the time. Also, inhalers can be found for cheap in a pharmacy. Its not necessary to schedule a doctor's appointment to get a prescription for one like many places in the US. So that's nice, saves some money and time. Anyway, this is all I can think of for this post. Hopefully, you will not need a hospital during your stay in Amman.