I was wondering if there are any good exercises or tips to help me get better/faster at alternate picking when it comes to gothenburg style riffs. I can down pick them decently but then my wrist and part of my arm starts to become tired after a while. Is there anything to help me improve this, or does it just take time?
mostly just time, try and incorporate alternate picking in all of your playing and soon enough you won't even have to think about it
well it may take time but it shouldn't make you ache.
just slighly move your hand/fingers just enough so you pluck the sring again on the other side.
you can also rest your hand on the bridge if you want.

to get used to it
play scales cromatics and your riffs to get used to it. you'll soon find it weird not to alternate pick