Best drug for acute gout in a patient with renal impairment is:
Naproxen: The anti-inflammatory activity is stronger and it is particularly potent in inhibiting leucocyte migration - may be more valuable in acute gout: dose 750 mg stat followed by 250 mg 8 hourly till attack subsides. It is also recommended for rheumatoid arthritis and ankylosing spondylitis. Because of longer t ½ , regular use can effectively suppress platelet function. Gastric bleeding is more common than with ibuprofen. Naproxen carries lower thrombotic risk than diclofenac, etoricoxib, etc. Dose should be reduced in the elderly.
Naproxen is marketed as active single S(-) enantiomer preparation, which poses less renal burden. However, some R(+) enantiomer is formed in vivo due to inversion.

Uricosuric drugs like probenecid and sulﬁnpyrazone are ineﬀective in the presence of renal  insuﬃciency.
Allopurinol is a drug of choice for most cases of chronic gout. Renal  impairment  increases  the  incidence  of rashes and other reactions to allopurinol.

​Reference: Essentials of Medical Pharmacology Eighth Edition KD  TRIPATHI  page no 217
Naproxen
Probenecid
Allopurinol
Sulfinpyrazone