Some private companies offer ~$80 rides directly from the airport to Jasper, while Greyhound charges ~$40 from the city (Edmonton) to Jasper. Currently it seems that the cheapest option would be a bus to the city and a bus (Greyhound) to the park. Any other options?
Via Rail has service from Edmonton to Jasper every Sunday, Tuesday and Friday, and return service every Monday, Wednesday and Saturday. The lowest fare I could find was $182.70 return. Probably not what you want.
Greyhound Canada has daily service (though it leaves Edmonton at 12:30 am!) and I was quoted a price of $72.70 return for a 14-day advance purchase on their web site. This is the best option I could find, though with that departure time you should probably plan to spend a few hours sightseeing in Edmonton.
Edmonton also has the "747" bus which runs between the airport and Century Park Station; this costs $5. You'll then need to take the 501 train ($3.20) to Bay Enterprise Square Station to get to Greyhound, which is a few blocks away from that station. The Via Rail train station is not very easily accessible via public transit, but it can be done with a bit of walking.
I wasn't able to find any other options cheaper than Greyhound, though there is always hitchhiking.
At $80 from the airport directly to Jasper vs $40+ from city center you value your time at nearly nothing. You will waste considerable time traveling to downtown, and then waiting until midnight for the bus. All for about $30.
Or you can land at the airport, get on a highway bus and be in Jasper in about 5 hours.
If you're travelling in a group, the cheapest way could be renting a car in Edmonton and returning it in Jasper.
Note that the one-way renting prices of Edmonton<->Jasper are much cheaper than Calgary<->Banff.