Simply put, it is illegal to walk along the railway line, or to cross it anywhere other than at designated points - and although there is traditional access to the hill by crossing the line near the bridge at the west end of Loch Eilt and just before the tunnels above the Muidhe (both points have access gates) I don't think even these are formally recognised as designated crossing points.
Therefore, the only proper ways to get to Essan is by helicopter (!), by boat - but only if you go under the railway bridge and moor up at the mouth of the Easain burn (not always possible when water levels are low) - or by walking in from Inverailort, over the hill.
The original track to Essan ran alongside the loch and was covered over when the railway line was built. The nearest to an 'official' legal route in today is from Glenshian Lodge and up along an argocat track which roughly follows the Allt an-t Sagairt up towards the NE ridge of Beinn Coire na Gall. From there, just drop straight down towards the bothy. It's not too bad a route in good weather. Or you can climb straight up onto Creagan an-t Sagairt and follow the ridge above Loch Eilt to Creag Dearg and then down. This does offer some good views. There is no path of any shape or form down to Essan from any direction.
When the water level is low it would be possible to walk most of the way in along the shore - can be a bit muddy in places though.
Other than that, all I'll say is that I reckon I know every sleeper along that track better than the pavement outside my front door