You have many
options with those parameters.
Southbank, at the end of North Bridge Road near Lavender MRT, is one example. At this instant there's an 82 square meter (large) one bedroom listed for S$3200/month, and the two bedroom apartments routinely list for S$4500 at this point in time. (I'd pick the large one bedroom in that case and conserve your budget.) You're three stops away from Raffles Place MRT (9 minutes aboard the train according to Google; I'd say 10 at rush hour), the Kallang River area is quite nice (with a great path for jogging along the river), and the immediate area has all the services you'd need, including a FairPrice supermarket just north of the MRT station. Arab Street is close by, too. Citylights is another example, on Jellicoe Road. I see a couple ~82 square meter two bedrooms listed there for S$4000 at this instant.
There are occasionally some nice HDB units that pop up on the rental market in the same area just north and adjacent to the MRT station, near Citylights. One of them I spotted a while ago even had a dishwasher (a rarity in Singapore). They would violate your gym/pool requirement, but I would still ask to take a look at a couple for reference. You'll be astounded at how much that gym and pool adds to the monthly rent, and then you can consider whether you'll just buy those particular amenities separately, at a fitness club near your office for example. Or just get a bigger sized HDB apartment to house your own private gym.
Tanjong Pagar, one stop to the west of Raffles Place MRT, is another possible area but probably a couple hundred more per month on average. You might stretch your budget there, but I don't think you'd break it if you shop around a bit. Tiong Bahru gets a bit more affordable, and that's a really interesting area with some nice markets. Tiong Bahru is also 3 stops away from Raffles Place MRT on the East-West line.
You could go as far north as Novena MRT and maintain that 20 minute commute, but for some reason the rents in that direction seem to be a bit higher. Maybe it's the proximity to Orchard Road.
Downtown Line Phase 2 is opening at the end of this year (2015), and there's a station called Downtown on that line that's very close to the CBD. (Hence the name.) If your office is near the Downtown MRT station then you could take a look at that line, particularly something near Rochor MRT station (4 stops away). However, there aren't too many apartments near the new Rochor MRT station. I remember looking in that general area, and I wasn't able to find much, but you can try.
Don't forget to look at the bus lines. If you've got a short bus ride, that still works very well.