There are trains from RMM to MDU.
Or you can can use TN govt buses from RMM to MDU. Lot of buses available.
At Madurai, not sure if the buses from Rameswaram goes to Mattuthavani or Arapalyam bus stand. Again not sure if buses to Kumily are from which bus stand.
But both bus stands have local bus connection.
Buses...
more... to Kumily are a lot through the day and night.
Madurai is a "city that never sleeps". Lot of buses in all directions and with a good frequency thru the day and night. Even local city buses.
At Kumily, the TN buses will stop at one side of the checkpost/border. The other side of the checkpost is the town of Kumily and KL bus stand.
Thekkady is few Kms from Kumily town. Accommodations are available at Kumily. Thekkady is a tourist spot and you can hire local vehicles to go there.