Typically cheap and longer range shooting do not go together. However an Apex or lower end Omega will get the job end. I have a higher end Omega I use for long range shooting (several deer over 300 yds) and even that I've done extensive work on to take equipment limitations out of the equation.
For the gun, I would get an Omega or Apex. Hope you get a good one because if it's not accurate with powerful charges it's not a 250 yd elk gun.
For scope, remember you're going to be talking around 24-26 inches of drop at 250 yds even for the flattest loads. I'm a fan of tactical scopes because you can dial in exact drops, but a good one will not fit your price range if you're looking at lower end Omegas (they start at around $400-$600). I would recommend either a BDC-reticle scope and accept that the marks will be off and you'll have to shoot to figure out where it REALLY shoots at 200-250yds, or get a Mil-Dot reticle and do the same thing.
For bullets, I recommend the .458" 325gr FTX. A big strong bullet that is also built to shoot flat. I drive it at 1950 fps with 135gr Triple Seven FFg. I get 26" of drop at 250 yds. There are other good elk bullets out there but most are not designed for as good of energy retention to 250 yds (ie high BC) and with elk that is very important.