Does a landlord suppose to give some type of compensation for your air conditioner now working to a whole week? And other there thing in the apartment??

The only time a landlord will give compenstation for repair issues is if you paid or the repairs out of your pocket. A landlord is reponsable for fixing things in a "reasonable amount of time" this can very by state. Check your lease agreement. If the a/c is still not fixed, you could hold rent in escrow until the problem is taken care of.