Have no experience in goats, but have gone through the same thing with aged sheep. Tried herbal supplements which I don't think helped, much as I wanted them too. Also tried Bute which did help for a while but was difficult to get into her - she managed to eat all her coarse mix and leave the powder behind. Ended up core-ing an apple, tipping the Bute in and putting the core back in. Fiddly and time consuming. It doesn't dissolve well in water so syringing down her throat wasn't much fun either. Then moved on to syringing dispersible aspirin which I do think helped. Problem there is we have large sheep so the dose was so big that it was nigh on impossible to buy enough aspirin. Each shop will only sell one packet! Maybe the willow is a good place to start.
With our last arthritic girl we went straight for Metacam and that did help her. It's not licensed for sheep, can have side effects from continued use and is awfully expensive!
Other things that helped - we put her and a friend in a stable overnight on a comfy straw bed to keep the damp away from her unless it was a nice warm night, and we had a coat made for her - like a horse rug with fleece on the inside and waterproof on the outside.
I hope this helps.