As stated by skinewmexico, Howard Roark, dmazur & browningguy, a properly set up .243 is an excellent choice for long range shooting, delivering better drop & wind figures than many cartridges "thought to be" more effective, with very mild recoil and good powder economy.
The limitations many people immediately think of are imposed by the use of very light bullets, which are indeed ineffective at longer ranges. The same problem can be seen in any caliber. A 50-55 grain .224 bullet vs a 77-80, a 140 grain 7mm, vs a 168/175, a 150 grain .30 bullet, vs a 175/190 . . . the list goes on for ways to hamstring long range performance.
A properly stabilized heavy bullet is a completely different animal and worth learning a little something about. At 1,000 yards, the 105 Scenar, launched at a moderate 3,000 fps, is still running 1,400fps +, and will easily stay supersonic beyond 1,200. Some folks, with longer barrels and more guts drive them substantially faster. Even with the 105 at 3,000 fps, a Federal FGMM .308/175 SMK goes subsonic several hundred yards before the .243 bullet.
The first key decision in a long range .243 is rifling twist, and a fast twist is needed to stabilize the long BHTPs like the Lapua and Berger 105s, the 107 SMKs and the DTACs.
I set up my daughter's rifle with a 24" 7.5:1 Schneider barrel as a low-recoil and high-efficiency, yet portable, long range match rifle. It does very well, and will shoot 105 Scenars into 3/8 MOA. With the 105s at close to 3,000 fps and the 115 DTACs at 2,925, the drop and drift figures rival my 6.5-284. She has no trouble keeping all of her shots within 3/4 moa at 1k, and if wind conditions permit, the rifle will actually shoot a little better than that.
Regarding the OPs original question, there are many scopes with will do a good job to 1k and beyond. Adjustable parallax, a 30mm tube and at least 10x would be my starting point.
With any true long range rifle, I'd suggest starting with a 20 moa base to give a head start on the elevation needed to dial on at these ranges. Laura's rifle needs 31.5 MOA from her 100 yard zero to be on at 1k, the FGMM .308/175s needs over 39 MOA, almost 6 FEET more - go figure.
For magnification, I would suggest at least a 4.5-14x, and a little more power is often an asset. I have a 4.5-14x40 on Laura's .243 and it fits the LTR style package well. A larger scope like a 6.5-20x would be right at home, as would be a NightForce 5.5-22x50.
More effective long range performance CAN be had by other cartridges, like the 6.5-284, the 7WSM (or 7x300WSM) or even a .300WinMag, but at the expense of more recoil and greater powder consumption, and in the case of the 6.5-284 and the WinMag, a longer action. The biggest advantage of the bigger calibers is stability in the wind, and downrange energy. We shoot a lot of matches at long range that use steel plate targets, and misses are much easier to spot (and correct) as bullet weight increases.