Akron airport is available via public transit, but by what will most likely be an annoying time consuming process. If you are interested, here goes.
First off, for your guidance, is the following website for Akron's bus service where you can schedules for all the routes and additional information:
From Akron Airport, take the 110 bus to downtown at Cascade Square. The fare is $1. You should probably tell the driver that you're connecting to the NorthCoast Express to ensure a good connection. You may have some time to kill in downtown Akron. The Customer Service area is on Main Place nearby in downtown.
From there, you take the X61 to downtown Cleveland. The number one issue with this bus is that IT ONLY RUNS ON WEEKDAYS. However, you get a straight shot from Akron the the Public Square, where you take the Red Line Rapid to Hopkins Airport, and then the LCT to Oberlin (see the Oberlin Transit home page for more details).
The whole region has just gotten so car dependant, so I apologize in advance. If your patience has already been tested just by reading the above suggestion, you can also take Hopkins Limosine (call 800.543.9912 for reservations), which operates a relatively new shuttle service from Akron to Cleveland Airport. From there, of course, you can take the LCT (or even connect to another Hopkins Limo going to Oberlin). The Akron/Cleveland limo, at the very least, is somewhere around 30 bucks one way. To get that quote you have to ride on their fixed schedule which may or may not work with yours.
Good Luck!