The solution is to just build the damn game.
Don't try to come up with some dumbass copout to explain his power levels, just let the game establish that his power levels are his power levels. Perfect example is how Superman:TAS and Justice League toned down his powers without any shitty explanations. Hulk: Ultimate Destruction did the same thing; they didn't have to rationalize in-game why guys with machine guns could whittle away Hulk's health, they just did it. Guys with guns shot at Hulk, it did damage, his character no sold the damage he took until he started to get hit with explosives, tanks, helicopters, and robots.
The solution for Superman is to do the same. Random guys with guns and regular humans do negligible damage and the character doesn't react --perfect for small crimes and whatnot-- and then ramp it up as he fights metahumans with decent power sets, enemies with some degree of super strength, robots, aliens, and whatnot.
Compensate for the fact that Superman is hard to damage by making him fight enemy swarms --Hulk: UD an Prototype did this well-- and then of course, mini-boss and boss enemies would be able to kick his ass solo, and having to fight multiple boss level characters would of course be hard.
Standard model of all powers unlocked from the start while still having the character level up should apply. You could rationalize it as Superman holding back less and less as the game goes on as super breath or heat vision are leveled up to do more damage and greater dps, or you could just not rationalize it because it's a video game.
Do a hybrid of open world and linear level design, with linear levels being things like Toyman's little whatever, Warworld, the Phantom Zone, and so on, and then come up with a plot that would prompt everything from regular guys to low to mid metahumans to team up to fight Superman along with random crimes and whatnot. Some sort of random enemy power mechanic should be involved so you won't just roll up to a bank and be like "oh, these red color dudes all can teleport", gotta have some surprise.
Vision powers would be the detective mode, hearing would act as an excuse for a map/radar/quest marker, and so on. Combat would be some sort of Hulk/Prototype/Bayonetta hybrid because Superman should be more than capable of transitioning from ground to air combat and back within the same combos. Regular human enemies would have their own attack animations where Supes does casual grabs and smacks because he's not going to haul ass and uppercut people to the stratosphere if they can't take it.
Other than this, standard superhero/action game fare should apply.
Superman games fail when they try to do the "he's invincible, so we got to build the game around that" or the "this character is too powerful, so we have to make a plot excuse why they're not in game and then wait until the final boss fight for them to be at full power." Superman isn't invincible; hit him hard enough and you will break his jaw. Hit him with magic and he'll stagger. Hit him with electricity and he'd writhe and scream in pain. these are all things enemies could be based around, and that's without Kryptonite.
Everything else, from Clark Kent segments, trips to the Fortress of Solitude, and so on aren't necessary. As long as the game is built first and power levels are reigned in in respect to those rather than building the game and narrative around power levels, you can't go wrong. You don't need a depowered Superman, a young Superman, an invincible Superman with the city as his health bar, you need the game first There's a reason why the best Superman game, even if it's still average as all get the hell out, didn't focus on any of that. They just built a game and stuck Superman in it.
