You will want to head around on I-83 (still US 322) to the interchange where everything changes on the east end of Harrisburg, locally known as the Eisenhower Interchange.
The shortest way to the south end of Lebanon is US 322. If you are on a normal day shift US 322 will stack up at the lights between I-83 and Hershey, and also passing the Hershey medical center and to some extent in Campbelltown (although you'll be reverse commuting from the Campbelltown perspective). Night shift should be OK.
A faster but more expensive alternative will be to continue on I-283 to the PA Turnpike, especially if you have EZPass. One secret to the EZPass is that other states may have lower or no extra monthly fees, but still works in PA.
Bad idea will be to take I-283 to PA 283, this expressway quickly goes to Lancaster which is out of your way.
A VERY bad idea will be to take US 422 from Hershey to Lebanon. Many, many, many traffic lights and very slow going.
I-81 north to somewhere isn't great either since it goes the wrong way then you have to work your way back through 1930's vintage roads through Annville and/or Lebanon City anyway.