You could get the bus or you could get a midland mainline train to Loughborough and then a bus from the station to EMA
. This is probably quicker, although be warned the AL1 bus only runs once an our from Loughborough train station to the airport.
As for what to do, go upstairs to the cafe, grab a coffee and watch the action out of the window (although there won't be much at that time). What i would say if you want to watch the action don't go airside until you have to as the domestic area is small, crowded and doesn't have views nearly as good as the landside cafe.
Enjoy your trip.