ISO 1600: 1/160th sec. @ f/5.6 = 1/640th sec. @ f/2.8 or 1/500th sec. @ f/2.8 at ISO: 1250.
This should get you the results that you are looking for as long as your original exposure is based on the players being correctly exposed and not the ice.
You should be able to find a used Nikkor AF 80-200mm f/2.8(D or non-D) ED One Ring (Push-Pull) for around your budget in excellent condition. Check www.keh. com and the used equipment at other well known Nikon Authorized Dealers.