You disembark from the Yellow Singapore bus at the Johor Barhu long distance bus terminal. The terminal is more what you might have expected Singapore's Queen St terminal to be, with ticket offices, shops, and food outlets. It is not necessary to pre-book onward travel from here, as the options are virtually unlimited. Continual buses depart from here, heading up both the east and west coasts of Malaysia, and even into Thailand. The local bus companies offer their destinations and departure times at their booking booths, so you can easily pick a suitable service onward. To look for bus dropping of at Mersing, look at booths booking the eastern coast destinations of Cherating and Kuala Terengganu.
We suggest it is a good idea to leave Singapore fairly early in the morning, so as to allow plenty of time to organize where you are going, and to catch a suitable bus, before it gets too late. There is nothing more inconvenient than to be trying to find accommodation in a transit stop, for no other reason that that time has got away. With border crossings and multiple buses this was always a possibility of delays beyond your control. However, leaving your Singapore hotel at eigthish, should get you to the Northern bus depot by mid day comfortably, allowing another hour or two for the bus ride to Mersing.
The national bus network does not appear to be used by many foreign tourists, and you see very few tourists at any of the bus depots around the country. This is really surprising given to low cost, and the easy access to so many places within Malaysia. Another benefit, is the busses are often quite empty, and this gives a bit of room to spread out on the longer journeys! The only tough part for the bus services, is knowing where they are going to drop you off. In the case of Mersing the bus drops passengers on the side of the through-town highway. A bit of a walk is then required to get accomodation etc.