I'm guessing that you are just assuming he has arthritis, and have not had that verified by a vet? If a vet did the diagnosing, he or she would also be the one prescribing treatment.
Antiinflammatories such as bute are prescription meds that are used to treat arthritis and they are not just about pain relief. They are given to reduce the inflammation which is damaging to the joints. As with any drug, long term use can lead to adverse effects, so veterinary monitoring for signs of those effects is a part of the treatment plan.
If your horse does have degenerative joint disease, which should be diagnosed by a vet, then various treatment options might be prescribed, including giving otc oral glucosamine and chondroitin preparations or oral MSM a try. These are generally regarded as safe if purchased from reputable dealers, but their efficacy is still a matter of anecdotal subjective evidence as opposed to objective scientific proof.
As one answer mentioned, you can ask a vet to prescribe Previcox, but you'd be asking the vet to violate federal drug law. Previcox is a firocoxib preparation that is not FDA approved for use in horses. There is an equine equivalent called Equioxx, so it is illegal for a vet in the US to prescribe Previcox to treat a horse. Vets can only prescribe off-label use of drugs when there is no equivalent FDA approved drug available. Firocoxib (Equioxx, or Previcox) given at a clinically effective dose of 0.1 mg/kg (.045 mg/lb) of body weight is recommended for no longer than 14 consecutive days of treatment in horses, and toxicity develops after 30 days of continued use, so Previcox is no miracle drug to replace bute. http://dailymed.nlm.nih.gov/dailymed/fda/fdaDrugXsl.cfm?id=18680&type=display
None of the prescription drugs should be given without ongoing veterinary monitoring, since all drugs have the potential to produce adverse reactions, toxicities and side effects, and not just bute. Those risks are weighed against the benefits when these drugs are prescribed.
Before just finding a drug to mask pain, your horse deserves and needs to be examined and diagnosed by a qualified veterinarian. No otc supplements, knockoff drugs, or homeopathic junk should be administered without veterinary evaluation and monitoring.