If you want good grass, you have to ‘feed’ the soil with organic ammendments to get fiber, particles, down into the soil. You can use you stall bedding material of sawdust, shredded leaves, floor sweeping from hay or straw storage. Any local farmers cleaning winter bedding from cow barns or shed that would spread it on your arena space? Then get the aerator to punch holes opening dirt to the organic material. I really would not add sand. It doesn’t loosen hard dirt without machinery to work the hard soil. Microbial animal life will help break down the organic material, pull it down into the soil which spreads dirt particle for better water absorbtion, building actual soil footing over time.
Items you spread need to be natural products mentioned, that break down in weather, not specifically labeled organic or certified organic, like yard mulch, wood chips, used stall bedding. Canadians use chips, shredded wood for footing in winter, wet locations, usually called ‘hog fuel’. It breaks down over time, may get muddy up there with so much water. Probably not be muddy in your drier location. Still helpful with other amendments in a thin covering layer as arena footing, aid grass growth as the season progresses. Grass roots need protection from hot sun, drying wind, horse hooves, which the continual addition of the amendments will provide. Regular mowing will help prevent seed set, which causes grass to go dormant, not grow again until fall. Mow high, at least 4 inches, 5 inches is better, to protect the roots, feed the plants.
I have greatly improved our pasture grasses using this method, along with getting the fertilizer recommended by soil testing put on yearly.
One other thing helps. Stand by the new arena area and tell some friends this will be your new DIRT arena! I swear the land hears you, so every green thing takes root in that area, producing a plant cover on the dirt. May not all be grass, but it sure tries to prevent any exposed dirt! Ha ha Our large sand arena now is covered with volunteer grasses, some weeds and laughs at damage from shod hooves. Everything growing is a very tough plant, stays green all season without watering or fertilizer. We do mow it as needed to stay about 5 inches, graze horses on it now.
Good luck with your project.