That lathe is great for starting out! Also, if he ever wants any upgrades or needs spare parts (drive belt for example), it's the exact same lathe as a Rockler Excelsior.
Sounds like you've already got those chisels, but to be honest the tangs look pretty small. They're probably okay but they seem like a (more expensive) knock off of these: https://www.amazon.com/PSI-Woodworking-LCHSS8-Chisel-8-Piece/dp/B000KI8CTS/ref=sr_1_4?dchild=1&keywords=PSI+lathe+tools&qid=1606248928&s=hi&sr=1-4
Turning pens is really fun, but there are some fiddly things he'd need to get going. You can get away without a press; most people without one just use a hand clamp (the kind where you squeeze a grip to close). He's probably already got those, and decent presses are pretty expensive. You definitely want a mandrel. That lathe has a #2 taper. I've got this one and its fine. Just make sure he doesn't over-tighten the tailstock and bend it, but it works with that lathe's stock live center. If he's already got a drill press then that's good, but if he's going to need to drill on the lathe, he'll need a jacobs chuck for the tailstock and a scroll chuck with jaws to hold the work. The jacobs chuck is cheap, but the other is pretty pricey.
For the pen kits themselves, you'll need the bushings for that specific pen kit. A normal slimline pen will use slimline bushings, but more specialty kits will have their own bushings. These go on either side of the blank on the mandrel, and give a reference so you know what size to make the ends so they fit together nicely. Just be sure he has the right bushings (and maybe some spare slimline bushings just to take up extra space on the mandrel; the brass knob holds everything tight, but the mandrel only has threads at the end. There are also some vinyl bushings to use while finishing - these prevent you from gluing the parts to the metal bushings. It'll still stick a little, but these are easier to pull off than a full metal one and you can just use a razor to trim off any excess dried glue from the ends.
Lastly, if he is going to do a CA finish on the pens, here are two tips that will save him a ton of frustration: after sanding to ~400 grit, be sure to use thin CA, and use an accelerator between coats. Then use some micro mesh sanding pads. I seriously can't recommend the accelerator enough; tried a bunch without it and it just takes so long to dry completely he'll definitely get impatient and it'll just gum up his sandpaper. And to get a super clear finish those micro mesh pads are amazing.
9 grits of micro mesh from 1500 to 12000. Like these: https://www.amazon.com/Micro-Mesh-Sanding-Pads-Grits/dp/B0756MT6X8/ or these: https://www.amazon.com/Sanding-Stopper-Polishing-Perfect-Materials/dp/B000H6EC4C/
I highly recommend using these. It takes more time than steel wool but works better and cleaner.