The cheapest way to Johor Bahru would be to take Bus No. 170 from either Queen Street Bus Station or the bus-stop along Rochor Canal Road (just after Sim Lim Square).
The bus fare is S$1.60. The service runs from 5.20 a.m. to 12.10 a.m. everyday.
The bus will stop at the Singapore checkpoint. The bus will not wait for you whilst you clear immigration, so take all your belongings with you. Keep your bus ticket and hop on the next Bus No. 170 which comes along. You don't have to pay again.
The bus will stop again at the Malaysian checkpoint, which is at the other end of the Causeway linking Singapore and Malaysia. You will have to get off to clear immigration. Once more, the bus will not stop for you to clear immigration, so keep your bus ticket if you intend to catch the next bus.
After clearing immigration, you have the option of either walking straight into Johor Bahru or to hop on the next Bus No. 170. The bus will take you to the bus-stop outside Komtar Shopping Centre (along Jalan Tun Abdul Razak) first, before stopping at Larkin Bus Terminal (at Jalan Datin Halimah in the outskirts of Johor Bahru).