How would that work with the electrical wiring running on the ceiling to the base?

turn off circuit breaker to light(s). remove lights, remove mounting fixture, disconnect electrical wires, install new lights, turn circuit breaker on.

1. Is there a standard electrical box (aka "j-box") inside the blue mounting fixture to which the existing light is attached?
2. What is the diameter of the blue mounting fixture?

How would that work with the electrical wiring running on the ceiling to the base?

Appears that the globe fixture is mounted to a Wiremold Solid Base Fixture Box Fitting.

https://www.legrand.us/wire-and-cable-management/raceway-and-cord-covers/steel/wiremold-500-700-series-solid-base-fixture-box-fitting-ivory/p/v5738a

The fitting is essentially a junction box (j box). Should be no problem to fit a surface mount fixture to. Just do as A Mat outlined.

That Wiremold box is 5-1/2" in diameter. This Progress Lighting 5-1/2" Emblem Collection Surface Mount LED is the same diameter so it might look the best. I'd paint the box to match the ceiling. An alternative: this Juno JSBC SlimBasics at 5-1/4" in diameter.

What you have there is a "HOMEAIDE LIGHT" seems that closet didnt have a light so they sucked off of a plug that was down low and added a flush mount. This means it probably has deep canopy with a whole drilled into the side to let the wires in. It also means this is just a closet, and probably replacing the light bulb with a bright white LED would solve the low light issue. alternatively you could put a short piece of track lighting and to track heads to light it up better and works will with the electrical set up.