That should be plenty of time, assuming your inbound flight is on time and assuming you are checking bags through.
Alaska arrives into terminal A and Aer Lingus departs from Terminal E. There is no airside connection from A to E, however. You will need to exit security, transfer terminals, then go through security again.
The quickest way to get from A to E is by walking, not by waiting for the shuttle bus. The best route is to walk through the Central Parking garage. When you exit security in Terminal A, look for signs for Central Parking. There is a skybridge over the roadway into the garage. When you are in the garage, look for signs for Terminal E. It is all very easy and obvious.