if using my own player might be considered “good enough,” couldn’t I just hook my Walkman’s headphone output to the line in or mic input on my computer and do it myself?

absolutely, i was responding to your question

any advice on what I should do to get the best quality transfer that I can?

other users have mentioned you can get a converter online, but the “you get what you pay for” maxim applies to electronics maybe more than anything else. the difference in quality between “consumer” and “professional” audio gear is getting narrower, but it’s still there-- everything just depends on how much you want to spend

not sure how I need to set the volume on the Walkman

if the walkman has a “line out” port, you’d use that to plug into your “line in” on your recording device. you can use the headphone jack, but that signal is already amplified, so you’ll have to adjust the output volume to where the input meter is the highest it can be without ever clipping (going ‘red’). older consumer gear will have more noise (hiss) than anything professional. especially an amplified signal, as in a headphone jack. and that’s where the money for pro stuff goes–lower noise floor. more information

edit: on a whim i did some looking and found this lol

you could just buy that, dump the recording down to a SD card, and then return the device the next day. done.
i had no idea they made such things, but i haven’t really kept up with the music/audio industry since i left