My apt is very old including all the water fixtures. I took the spout to my kitchen faucet off and it is full of rust really bad. My question is if my water is traveling through all this rust is my drinking water unsafe?

Most all of the rust clings to the iron and is only slightly dissolved in the water, it's safe. How do your white closes come out of the washer?

If you are really concerned about it, you should take a sample from the water out of the fuacet and have it tested. This way you will know for sure whether it is safe to drink or not. Please let us know how the tests come back.