by my experience with visitors , we always face how to make a plan to see/do maximum , in minimum time.
So I’ll suggest you this places deepened on you available time :
Time 4 hours or less .
Place : Amman
1- If you like to see panoramic Amman , and feel you up the world, you just have to visit Jabal al-Qal'a, (Amman Citadel) from Neolithic period,.. you can see all area in Amman ; downtown , The Roman amphitheater in Amman, Jabal Amman , Royal palace ,and the great Jordanian flag, it’s open till 7 pm.
2- Kahf Arqim (The Cave of the Seven Sleepers)
in suburb of Amman called " Araqim" or "Abo Alanda". There is a cave , that people it's a cave of 7 sleepers. It's beautiful place, not far from city center , maybe 13 Km (not more 5 JD by Taxi).. There is many gardens with smile of Lavenders and Roses.. Beautiful not tall palms , and many Olive and Pine trees.. The story of this sleepers was written in Quran.
open to 7 pm
3- JABAL AMMAN and Rainbow street.
A good place to take the pulse of the neighborhood is along cobblestone-clad Rainbow Street – the road signs used to say Abu Bakr Al-Sideeq Street, until the residents voted to change the official name to match the one everyone calls it by, after the Rainbow Cinema.
Coffee shops and restaurants work after med night.
4- If you like shopping let me suggest visit great and beautiful mall called Mecca mall.. many brands name , many shops, tasty food corner from around the world cousins. And grand cinemas in 5th floor.
Or you have to go to “city mall” especially for daily shopping at Carrefour.
There is new malls like “albarakah mall” avenue mall” im Swaifiah , or Taj mall, abdoun mall in Abdoun area, there is many street shops too and 5starts restaurants.u feel morden amman life style.
5- Downtown … it’s great to see all Amman in one place, you can meet history from The Roman amphitheater to Islamic era , Alhussainy Masijd (mosque) ,and meet all kind of people , ritches and poor, conservatives women and eastern life style women, tourist and local Bedouins, young and older people, blonds and Naggers . In downtown market you find all good u dreams about, and in reasonable prices.