Yes, you can go from downtown Vancouver to Lynn Canyon Park by public transportation without any problem.
To get started, you must go to the Waterfront Station, located on Vancouver Harbour, very close to Canada Place.
At the Waterfront Station, you must purchase DayPass tickets. A DayPass provides unlimited transit use on all buses, SkyTrain and SeaBus for one day. You can use it for travel through all zones, and save money over buying single fares when you take multiple trips in the same day, individual tickets are more expensive.
Once you have purchased the DayPass tickets, follow the signs towards the SeaBus and towards Lonsdale Quay ( North Vancouver ). You must continue down the walkway and down the escalators to reach the SeaBus loading area. The SeaBus is a passenger-only ferry without outdoor seats, but if you pick a seat near a window, you can enjoy great views. It crosses the Burrard Inlet, connecting Downtown Vancouver with the North Shore and departs every 15 minutes during the day.
Exit the SeaBus at the Lonsdale Terminal, walk across the street and board bus 228 at Bay 5 towards Lynn Valley. After about half an hour of travel you should get off the bus at stop 54185 for Lynn Valley Rd at Peters Rd. To get to the Lynn Canyon Park entrance you must walk 0.6 miles ( 1Km ).
The access to the Lynn Canyon Park and the Lynn Canyon suspension Bridge is free. The cost of all the transportation for two it's about half the cost of a single admission ticket to Capilano suspension Bridge !!!
Crossing the Lynn Canyon suspension Bridge is fun, but sometimes the park can be very busy and unfortunately is really difficult to stop and enjoy the moment, so get there early !
One day is enough to visit the entire Lynn Canyon Park.