You've got a couple of options.

THWN Conductors

You could simply run THWN conductors inside the lamp post.

Underground Feeder (Type UF) Cable

You could also run a smaller gauge UF cable inside the lamp post.

In either case, you'll want the conduit to come up above ground. Either inside the lamp post, or next to it, depending on the post.

If the splices are underground, you'll want to use a splice method approved for wet locations. Conduit underground is considered a wet location. Even though the splices may be in a conduit body, it's still considered a wet location.

If the lamp posts have access panels built in to them, you can make your splices inside the lamp post. This would eliminate the need for a handhole or conduit body.

You CANNOT move the overhead line to underground. Everything from the enclosures, attachments, and the cable itself are different. Basically you'd have to remove the overhead line and associated parts and re-do the run with conduit (or direct bury). Where or not you can use the existing panels and such would need to be determined on site.

In my strong opinion this is not something for a casual DIYer. This is only something for a someone with a good bit of experience in this type of work.

With only knowing you will be running a welder it is impossible to accurately say what wire to use. Personally, with a home shop you'd almost certainly be fine with a 90-100A feeder. You can use 100A MHF cable (mobile home feeder) which is readily available anywhere. IMO anything smaller isn't worth it in the long run. One thing, due to the length of the run, you may want to drop the feeder breaker to 70A to curb voltage drop. Again, this all depends on what will be used at the same time.