How would someone with electric superpowers make a railgun?

He would be able to make one, and he could fire it, but there would be problems.

You see, even though he could shoot it, it would be a one-shot emergency weapon, as the recoil would be truly tremendous. As in "firing while standing equals broken wrist / shoulder" stupendous. It would be like an eight-year-old firing a 12 gauge shotgun (trust me, it's not pretty).

Also, it's worth mentioning that even if the recoil doesn't knock him into next week the gun would still be a 1 (possibly 2) shot weapon; one of the main reasons why railguns aren't in use today is because the conductive rails wear out extremely quickly.

Kinda obvious problem: we can't actually do a very good job of making a rail gun right now, even with access to plenty of money, power, engineering talent and high quality, high precision manufacuring. Unless your dude has super machinist powers too, any weapon he makes will likely be a less effective projectile thrower than a gun that he could probably just buy.

Lets see what we can do, though.

Lets say that to start with, we want something that is at least as capable as a regular handgun. Our theoretical handgun fires a 10g projectile at about 400m/s (compare with a 9mm parabellum, which is more like 8g and 380m/s). You want an "arm cannon", I'm not sure how big that is, but lets say it has a 25cm barrel (so, a forearm cannon, really). To reach the desired speed, you need an acceleration of 3.2x10⁵m/s² (about 2/3rds of the acceleration the 9mm bullet gets). On a 10g bullet, this needs a force of 3.2kN.

A railgun works by having a magnetic field pointing in one direction, an electric field at rightangles to that, and there interaction producing a force in the projectile at rightangles to them both. The pair of rails point at the target, the projectile sits between them, and you whack a massive current across the two (note that I said current, not voltage). You also need a load of magnets above and below the rails to provide the magnetic field. These will probably be electomagnets, and you'll need to feed them with a generous amount of current, too.

The simplest (and wrong, but it'll do here) model of a railgun says that $F = Iell times B$, or force is current multiplied by the length of the barrel multiplied by the strength of the magnetic field. An iron core electromagnet can have a maximum field strength of 1.6 tesla, though alternative materials can push thus up a bit. A real world example of a suitable magnet would perhaps be GMW 3470, which is about 38*28*23cm, and weighs over 30 kilos. It can develop over 1.6T and only costs a few thousand dollars. It is a very high grade bit of scientific kit, so maybe a more industrial one will be a bit cheaper and lighter, but I bet the figures are in the right ballpark. I'm not sure if you'll be able to run it at a high enough power level for a railgun magnet. It also says it needs water cooling if operated at high power. It isn't entirely adequate, as you'd be hard pressed to fit enough of them into your barrel space to provide a uniform magnetic field of high enough strength. You can of course use lighter, more compact, less powerful magnets, but you'll need to increase the current to compensate. Really, you need some room-temperature superconductors, and that's a whole new set of superpowers.

To provide the required force over the given length, we need a current of 8kA (more than 20-40 times more current than a powerful portable MIG welding rig). The projectile comes out with 800 joules of kinetic energy. It takes 1/800th of a second to leave the barrel, meaning that your gun must develop a power of 640kW whilst operating, if it is 100% efficient, which it won't be. Wastage will appear as heat, though I won't work out how much right now.

Even with this teeny tiny toy railgun, you can see that the power requirements are Quite Demanding. Sure, all those kilowatts and kiloamps only have to be supplied for a very brief length of time, but switching all that on and off precisely and avoiding nasty ringing effects (the magnets discharging might shock your dude, which could be embarassing) is quite challenging, which is why we don't have any railguns right now. Your rails need to be machined to very high tolerances, because they must contact the projectile perfectly, without airgaps because that will result in arcing which will waste power and damage the railgun. Your bullet will wear down the rails a little anyway, which will result in arcing soon enough. If the bullet gets stuck, it risks being welded to the rails immediately ruining the gun. The rails must be very firmly braced to prevent reaction forces bending them, which will cause rail-bullet wear or separation, resulting in more arcing. An 8kA arc is going to toast stuff pretty quickly, which is why current real-world railgun designs need to regularly swap out the rails (which is why you need super-machinist powers, to make lots of high quality replacements).

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Note that the average power required for the railgun (again, at an implausible 100% efficiency) is 800W, assuming a rate of fire of one round per second (it'll scale linearly with the fire rate). That's a simliar power output to a serious aerobic athlete. I don't know how you plan to power your powers, but if you want to tend towards realism, he'll need to eat well and he'll get tired quickly in combat, especially he's doing anything more than just standing there shooting.

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Lets look at the alternative. Use a common-or-garden, off the shelf handgun, if available, otherwise a rifle or a shotgun should be slightly more easily obtained. Just shoot people. Cheap, easy, all the machining issues are solved for you and mass production means costs are low. They don't weigh tens of kilograms and need no water cooling. If it breaks or you lose it you can always buy or steal another. You want to use electrical weapons? make like a taser and use your hands, and perhaps some weighed throwing wires.