There will be one or two Government buses right outside Mandapam railway station coinciding with the arrival of regular trains. Else you will need walk upto the town bus stop which is 1 km away from where buses towards RMM will be available at regular frequency.
Recommend getting down at Ramanathapuram (similarly the bus station/depot is situated on the Main road right outside the railway station) where you will have no issues getting a seat.
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Traveling time should be an hour from Ramnad to Rameswaram, while it takes 30 minutes from Mandapam.