The 70-200 VR used is older but fast focusing and quite sharp, well-made pro lens for $450-550 would be a great lens for arenas. I had a ranch were we breed and trained cutters, and light is always a problem. None of the suggested variable aperture lenses are suitable or any better in sharpness than your 18-55.
The suggestion of two primes, like the Nikon 85 1.8G for $550 is sharper and more sensitive in low light than any other lenses mentioned. It might be long enough plus makes a great portrait lens(so is the Nikon 70-200 2.8 VR, VR2 and P versions. For sports and fast moving action it also focuses faster than the competition.
If you got a fast 85 1.8G prime lens, you will need a wider angle also, the 24 1.8 was mentioned and is good but maybe not wide enough. The 20mm 1.8 is good for low light plus has excellent optical performance. If your were doing more landscapes you will need wider than 20mm. There are several 10-20mm lenses on the market. The Sigma 10-20 comes in a f/3.5 version and a lower cost variable aperture 3.5-5.6 that is popular low cost and very versatile which goes for about $200-250 used.Nikon came out with their own which is pretty good, smaller, for $306 at B&H So with the used VR1 70-200 2.8 you would have sports covered and landscape covered with the $306 Nikon 10-20 but that leaves a hole in the middle, so the 50 mm 1.8G for $200 from Nikon or for a real bargain, a Yongnuo 50mm f/1.8 AF for $60-70. Each of these would be a big step up from what you have with your light levels.