To reach the peninsula, there is a ferry from Breivikeidet in the borough of Tromsø to Svensby. The drive from Tromsø to Breivikeidet is about 50 minutes. From Svensby, there is 25 minutes' drive to Lyngseidet. From Lyngseidet there is a ferry onwards to Olderdalen on the east side of Lyngen. (http://www.bjorklid.no) for updated info.
Alternatively you can take the longer land-only route, following the E8-E6-Rv868 all the way round Lyngenfjord. This is the route to take if you're too early or late for the ferry.
There are a couple of buses a day from Tromsø to places like Svensby, Lyngseidet and Furuflaten. The northern tip of the peninsula is served by a bus a day, approximately, and a catamaran a couple of times a week. (http://www.177troms.no) for bus timetables.
