I had the standard '79
K model. I owned it for six years. I modified it for touring two-up (custom seat, saddlebags, fairing lowers, additional lights for conspicuity), installed an oil cooler (oil runs at 250 F, thermal limit of conventional oils, back in the day, not sure about modern conventional oils' thermal breakdown limits). Only reason I sold it is because we (wife and I) bought a dedicated touring machine: bigger, more storage, more comfort, more power, electronic cruise control, etc. My K was dry rotting in my garage, and I couldn't let that happen -- it was too good of a bike.
With regards to issues, the only one I know of is the head gasket can leak, very common. I had mine replaced, and it ran perfectly afterwards, no leaks. We rode it two-up, 1,935 miles, circling Lake Superior; ran it four days in the rain, in Canada -- never missed a beat, ran perfectly the entire time, even though we had it overloaded with us, and our gear.
Valve shims need checking/replacement periodically, but this is routine maintenance, on many bikes, old and new alike.
The Anniversary Edition is only slightly different than the K model, with mag rims, instead of spokes, and different badging. It might have dual front disc brakes, and maybe a rear disc brake, but otherwise, they are pretty much the same bike.
If it has pod air filters, ask if they have the OEM air box -- pods are extremely difficult to make work on the Constant Velocity carb's on these DOHC bikes. Cheers!
1979 CB750K (sold, 2012, but not forgotten)
1983 Kawasaki 440 LTD Belt Drive (sold, 2011)
1993 Kawasaki Voyager XII