Since 2015 KTM has run a commuter train between Johor Bahru and Singapore. The trains run from JB Sentral to Woodlands in Singapore (and the reverse). The cost is much higher from Singapore to JB. For whatever reason that is how the buses are priced too, the price is for example 5 MYR from JB to Singapore and Sing$5 from Singapore to JB. Since 1 Singapore dollar is 3.1 MYR that makes the price much higher from Singapore.
The train ride itself takes under 10 minutes. The schedule is subject to change but runs more frequently in the morning (from JB) and in the evening (from Singapore) and infrequently during the day. It is geared toward commuters from Johor Bahru to jobs in Singapore which is why the times may seem a bit odd at first glance.
Remember you have to pass through customs before getting on the train. They have made the process faster by having you go through both country’s customs office before boarding the train. With the bus you have to go through customs, go over the bridge, then get off the bus and go through customs again and then get back on the bus. This obviously wastes time. The JB extension to the Singapore MRT (whenever it finally gets started and then completed) will use this improved process of clearly both customs at the same time.
Frankly I find the KTM website too confusing to use. You may also purchase tickets via this site. Even on this site you have to be careful to get the right ticket (the results will include some that are not going between JB and Singapore – maybe that will be fixed when you look, but be careful).
The current schedule is (remember this may well change so try the links above for current information)