I'm Malaysian, based in KL, but have travelled a LOT to JB and Singapore, and probably the most unsatisfying and expensive propition, time-wise and money-wise, is doing the trip in either direction by air - contrary to what you might think, it does NOT really save time.
The one-hour bus ride from/to LCCT or KLIA itself is laden with irritation, cheap as it might be (approx RM10 only) - figure in an extra 15 minutes to 30 minutes' waiting in less-than-comfortable surroundings, and suddenly air travel doesn't sound so "fast" any more.
For best comfort and adherence to a schedule, take First Coach (from Novena Sq) or Aeroline, or even Causeway Link (Jurong East). Please note, do NOT consider any other lines, as reliability and safety are not priorities with them! You will arrive at your destination well-rested and cool 5 hours later, give or take 15 minutes.
If you're doing the bus trip from JB, use ONLY Causeway Link, and none of any of the other fancy-named lines AND BUY YOUR TICKET ONLY FROM THE Causeway Link COUNTER AT LARKIN BUS TERMINAL (not far from the McD's there). Ignore the "helpful" touts you will meet at every corner in the complex!
Train?! Forget it - it's noisy and less comfortable than the bus, and you likely will not get "a good night's sleep" in your 10pm - 7:30am trip, only to be assaulted by cut-throat taxi drivers as soon as you alight!! And believe me, they treat Malaysians every bit as badly as they would treat a Singaporean!!! Ha, ha, ha!
Good luck, and have a nice, pleasant journey.