The base will need to be underground to provide protection from radiation. Even just being in Earth orbit is a major radiation hazard when the solar wind is heavy. The reason we do not fry here on the ground is because the Earth's magnetic field and its atmosphere protect us.
These pretty lights are where the atmosphere blocks particles that would otherwise kill you through severe radiation poisoning
Your asteroid has no atmosphere and no magnetic field worth speaking of, and it goes into the very "stormy" regions of the solar system, right next to the Sun. By the look of that diagram, it even passes close enough to touch the corona.
This means the radiation hazard is extreme. If this base is sweeping that close to the Sun, they will need very heavy radiation shields.
Best way to do that would be to 1) bore down into the asteroid, since stone and metal-rich minerals protect very well, and 2) place the water tanks between themselves and the Sun. Water is an excellent radiation shield.
The major danger of taking a swim in a nuclear fuel pool is that the guards will be very cross with you for contaminating the ultra-pure water in the pool, you filthy thing you
So the main distinguishing feature of your base is that it needs to be underground.
All work sites related to this base will need to be underground as well, or be shielded, or have radiation hardened bunkers nearby where people can go when things get "hot".
This was a concern on the Apollo missions when — on the later, long duration J-type missions — the astronauts went for long treks on the Lunar Rover. NASA was very nervously watching the Sun for activity, because if the Sun had flared, Mission Control would have needed to recall the astronauts and have them take shelter in the Lunar Module to protect them from the resulting radiation.
With this I would expect the base to have one or several dedicated tunnel boring machines in operation.
Fusion reactors. Someone else suggested solar panels but I digress on that. Solar panels are extremely heavy compared to the power they give, and when you travel in space, you want to keep things as light as possible compared to the amount of "usefulness" they provide. If your heroes are "far more advanced" than us, then just assume they have mastered fusion instead. Energy is probably a non-issue.
Bring your handy fusion reactor, and energy will not be an issue. Plus you have a very sci-fi looking element of your futuristic base. (Source)
This will need to be brought, or — if they bring an extra fusion reactor — they can break apart materials on the asteroid to form these on site. Hydrogen is everywhere in the universe, and so is Oxygen. You just need energy to break apart minerals and molecules to form the right stuff.
After the edit to the question: If there is ice on the asteroid, this is a non-issue. Use the fusion reactors to melt and purify the ice for water; use electrolysis to make Oxygen from the water.
Purification and re-circulation of waste water / waste air will probably be necessary.
Expect that to need to be stockpiled. Unless of course — like air and water — you expect to be able to manufacture that on site. But unless your heroes have Star Trek type replicators it will probably be quite boring food... rather simple nutritional pastes and fluids, simple sugars.
Do note that this makes for a nice plot element when the transport from Earth comes with the good stuff... steak, juice, drinks. Also note that during the perihelion (passage closest to the Sun), no transports can come... the radiation is too heavy. And there is probably very little work to do during perihelion since they have to huddle in the radiation proofed areas of the base, sucking on their protein pastes and mineral-supplements.