The MGT shuttle was $7 per person each way last year. Rumor is it's going up a buck this year, but we won't know for another few weeks.
This is NOT the city bus... the city bus has its nearest stop just over a mile from Mendenhall Glacier.
Also, to the OP: The West Glacier Trail is much more difficult to access, as it's on the other side of the lake and river... I'd guess 3-5 miles on the back-loop highway from the visitor center on the East Side.
Other ways to get to either trailhead: taxi will run $30 (ish) which isn't bad if you've got a group of 4 or more. Some excursion vendors (whale watching, sport fishing, etc.) can drop you at the glacier after a tour and save you some shuttle time.
There are also several great hikes around downtown. Perseverance trail, the flume, Mt. Roberts, Sheep Creek and Pt. Bishop are all within a walk or very short cab ride of the ship terminals...
Better not for now... signatures get me banned!