It's possible, and quite a few people do it, but I would not recommend it. Singapore Expats has a good thread on this, but I'll summarize.
There are only two ways to travel between Singapore and JB:
- The Causeway from Woodlands to central JB. This is perennially congested, and central JB is (IMHO) not a particularly pleasant place to live.
- The Second Link from Jurong to Nusajaya/Iskandar and the expressway to KL. Much less crowded, although still packed at peak periods, and the Nusajaya area is fairly nice, although still under heavy construction.
Your main problem is going to be that public transit across these is pretty woeful. Buses across the Causeway are plentiful, but interminably slow, plus you need to disembark at both ends to go through Customs. Trains across the Causeway avoid the jams, but are very infrequent/unreliable and terminate in Woodlands, from where it's a bus plus a long MRT ride to Dhoby Ghaut. In addition, Immigration/Customs on both sides can be quite slow.
If you are going to do this, you'll want a car and cross the Second Link. However, the tolls will add up very quickly, driving Malaysian-registered cars in Singapore is (intentionally) very expensive, and parking around Dhoby Ghaut sounds expensive too.
The vehicle of choice for locals doing the commute is a motorbike: tolls are cheaper, jams can mostly be avoided, and parking is easier. However, I would consider biking in either country to be pretty close to Russian roulette. (YMMV.)
Things may improve c.
2019 2022 2024 2026 if they ever get around to building the Singapore-JB RTS, but at time of update (2020) they haven't even started.
And the final fly in the ointment is that you will also need to work out how you can stay in Malaysia long-term legally: Singaporean status won't help and a tourist visa won't cut it.