I don't want to get into a debate as to which is the fastest/cheapest/best way to get to John O' Groats, but would point out that some of the info offered on this thread is outdated.

There should be no great problem cycling from the airport to Inverness railway staion, approx 8 miles on level terrain although the road can be quite busy at morning and evening rush hours. Although there is a saying in these parts that flying is OK only if you've got the time to spare.
There are three trains daily from Inverness to Wick, if you are unable to book a bike space on any of these, then consider taking one of the six trains a day which take you as far as Tain or Invergordon. Scotrail do not take cycle bookings for these shorter hops, but apart from the connection leaving Inverness around 18.00, I have never seen a cyclist refused.
I believe Scotrail no longer transport cycles by van, but I think they only ever did that in the southbound direction (the van carried a load of empty coffins northbound

).
Have you considered the rail sleeper service north to Inverness? Always room for bikes in the guards van, and very cheap if you book in advance, look for Bargain Berths on the Scotrail website. Even at the full fare, it is not so expensive if you think of it as B&B on wheels

.It gets in to Inverness at 08.30.
Another option to consider north of Inverness is the bike bus run by Tim Dearman
http://www.timdearmancoaches.co.uk/durness.htm