There are no Apple stores in Thailand although there are unofficial resellers and most will send them to Chonburi for repair
For repairs in Bangkok you can go to Mahboonkrong Shopping Centre, Level 5, section 2. There are plenty of shops there that will do a replacement screen. Go to Siam Square on the MTR overhead railway and there is entry to Mahboonkrong from the station. This is the easiest to get to unless you are staying in Bang Pa when Pantip Plaze will be closer, it is opposite the Thao Maha Phrom Shrine which is opposite the Grand Hyatt Hotel.
Also in Bangkok there is a shop in Patpong Soi 1 off Silom Road next door to Kings Club. Get a Tuk Tuk from anywhere in Bangkok and just say 'Patpong'.
In Chonburii there is an authorised Apple repair centre but a smashed screen can take up to five weeks to repair. The shop is in The Avenue Chombrii (city) but is difficult to get to.
In Chonburi (province) the best place to go is Tukcom, 4th floor, South Pattaya Road, Pattaya Tai, Pattaya If you ask for Wat Chai Mongkol (temple) Tukcom is on the opposite side of the road about 100 yards walk.
In North Pattaya you can go to Big C Superstore 2nd Floor, next to Alcazar Transvestite Cabaret, get a Baht Bus from Royal Garden Plaza and stay on until the bus terminal. Big C is on your left.
My personal preference would be either Tukcom or Mahboonkrong, you will be able to haggle the price down to about 50% of what they first ask, it is expected that you haggle.
