The short answer is probably! How much, and how soon depends on your skin type. Some people are lucky, and don't go green unless they wear copper for days, others find it happen within hours. It all depends on the amount of moisture the copper is in contact with, so removing rings etc before you wash is a big help.

Is there any thing I can do to prevent this?

There are a couple of easy solutions! To form a barrier between your skin and the copper,
coat the inside of the piece with clear nail varnish or bees wax.
I go for clear nail varnish, as its long lasting, and you may already have some. Beeswax is a natural alternative,
but needs to be reapplied when it rubs off.
I don't coat my jewellery in advance, as people often want to absorb the holistic properties of copper :)

Is Copper good for Arthritus?

As far as I can tell, there is no medical evidence that copper helps with arthritus or rumatism. However, there are litterally hundreds of years of anicdotal evidence that suggeststs that it does. The Romans belileved that it aleviated the pain from swollen joints, and it had been a strongly held belief ever  since. And if it was good enough for the Romans, thats good enough for me!
Many people swear by it and will tell you theyve found great relief from wearing copper.
I dont know if thats a placaebo effect, or something that we cannot measure - but if the end result is painlessness, Who cares how we get there!
I'll leave it up to you to decide what you think is true.