There's a few (somewhat) simpler builds for it. It just depends on the run and the amount of levels you gain (exp wise) before the Arena. It tends to show up on level 2-3, so expect to be about experience level 2-3 when you go in. Horde every shotgun/shell you can (weapon switching and backpack grabbing is quicker than reloading with many builds). If you can get it, having a chaingun already makes things a bit easier for you. These builds do assume that you're going for the badge, not trying to complete the game with a super build.
Fin->Jug->(EE)? is a good build, providing easy weapon switching. Shotgun out the daemons/caco's with a pistol/chaingun for backup if the daemons get way too close. Switch to the pistol/chaingun after the barons come out (shotties really don't go well against their armour). If you're a marine, chucking BA on as your last trait (assuming you actually finish the Arena) is a nice little bonus so that you don't waste a single bit of the supercharge.
SoaG->SoaG->(SoaG/DG)? works, but is still a little too slow/weak for my liking. You can do it with a pistol build, especially if you manage to connect with radar shooting a little(see above comments for watching for ammo pickups by barons), but it isn't great. If you start as a technician (esp on AoMr), you can mod up one or two quite nice pistols from the get-go.
EE->EE->(SoaB)? Just a nice and accurate build. Radar shoot with shotguns still, but switch out to your pistol/chaingun as soon as you see them/know where they are. Cover is still your friend, and you'll be accurate enough to still hit a fair bit whilst running (auto-dodge is great).
Int->Int->(whatever/EE/Rel)? Scout-corner-shooting-madness. You always know when you'll hit with your shotguns, you can keep the hell away from things if you want, and you can run quite nicely. Probably one of the easier ones. Just horde shotties, shells and then blast the asterix out of, errm, asterixes. Chainguns will have an extra 50% miss rate if you can't see them(f'ing asterixes), so radar shoot with shotguns and slowly chip away at their health.
EE->SoaG->SoaG. If you get a chaingun before the Arena (chances are you probably will on UV), this makes it both fairly accurate and quite damaging.
There's a few little quick builds. I've done it with them in the new version (and failed to, I'm not that good at this game), so you can too.
Hell, people have done it with melee weapons on a regular basis.