I lived in Singapore for 2 years. The country is so small there is really no need for a private driver. Because of the small size and the large number of people cars and licenses are very expensive in Singapore. As such private drivers and even rental cars are pricey. They have an EXCELLENT subway system and cheap taxis though. Beware of using cars around rush hour as Singapore traffic can look more like a parking lot than a road at those times.
The subway will take you to all the main areas like, orchard road, and Chinatown easily, quickly, and cheaply. If you need to go somewhere special hop in a cab and pay the $5-10 or so. It will be slightly more if you are driving across the island but not cost prohibitive. The good thing is most of the main attractions are fairly close.
Depending on what time your ship arrives I would recommend taking a bumboat ride around Singapore, this will help you get situated and its fun to boot. Afterward you can stop off at boat quay for some food and drinks or across the river from it are some museums. Then head to Chinatown and view the beautiful buildings and the excellent Chinatown Heritage Centre. The heritage center is not to be missed in my opinion. Orchard road is great if you want to do some name brand shopping.
If you have to have a car and driver for other reasons such as limited mobility or large amounts of luggage let me know. I can put you in touch with some companies. Also if you do decide you need a car it helps to know a bit more when recommending a company. When does your ship arrive and how many hours do you need one for? (Some companies are cheaper for certain amounts of hours or don’t pick up at certain times) I take it you would like to be picked up at the dock and the dropped at the airport at the end of your sightseeing?