Probably JV level. Good soldering skills are important of course as are good desoldering skills since it's easy to damage the boards if you manhandle them.
As a high school and college student (late 70's), I did quite a bit of soldering and board rework as summer jobs and kept my skills up with car stereo installs and whatever household needs came along, so I have confidence in my skills in this area. That said, I knew that a bulb solder sucker and some solder-wick was going to make it more difficult to do well and take longer, so I invested in a Hakko soldering iron and desoldering gun last year and used them both on a refurb of my SA-9500ii. Since I plan to refurb units as a hobby, I thought the investment was warranted and after the 9500, was glad I did.
As for your 30 year old degree, well this gear is older still so you're probably okay. I have an electrical engineering degree, but spent my career in software so never really used it, though my being able to read and (largely) understand a schematic really helps when the inevitable troubleshooting is needed.
The SX-737 board with the volume control is tricky to get out without disconnecting some wiring, but it can be done. Based on the advice on AK, I only touched the caps on the tuner board since you may throw the tuner out of alignment when transistors are changed. Everything else was pretty straightforward.
Let me just parrot what you will find all over AK - take your time, read all the posts you can, get the service manual, build and use a DBT, and if you're not absolutely sure about something, ask.