Well, it's a tough time to arrive, that's for sure! There are western-oriented places that can be visited during the day, but you'll have to remember that there is no eating, drinking, smoking or gum-chewing allowed while out during the day. That said, here are a few places I know that are open at least for a little while, each day and then also in the evenings.
Carrefour - 9 a.m. - 1 a.m. Sat-Thurs - Villagio & City Center
MegaMart- 8:30 a.m. - 1 a.m. Sat-Thurs - Landmark, The Centre
Just about all the malls are open from 10 a.m. to around 1 or 2 and then re-open around 8 in the evening, so you can get a few hours of shopping in during the day and then go back at night if you run out of time.
I understand the Iftar buffets are marevelous and served at most hotels. That would be your late evening meal, and well worth the wait from what I'm told (I've only been here since late July, so I haven't had a chance to do all the Ramadan things offered yet)

Find a reliable taxi driver, and you'll be all set to pop around to the various malls to get better acclimated to the area. Once the heat eases off in a couple months, it would be easier to get around by getting a drop off at a central location and then walking around the city a bit. Get a Marhaba at a bookstore if you don't already have one; it is THE best resource for all this kind of stuff!

Have fun!