Lydia - i did my first half last weekend, and found drinking a bit of an issue. Advice i'd been given before ( but didn't follow very well ) was to slow right down at water stations to drink and maybe even walk. I tried to drink on the run ( plastic cups ) and got it everywhere except down my throat. By walking, you can drink properly without spluttering / half drowning yourself, and you won't lose that much time. To be honest by the time i'd gone through my last stop (which i walked through .. hallelujah !!! ) i was thinking it was maybe quicker to walk than run through and having 4 or 5 attempts to get it down !!!
If you do insist on running through, some other advice i've since been given that i will bear in mind in future is to squeeze the opening of the cup so it's smaller and doesn't spill as much.
Re how much to drink, everyone will be different but i reckon i drank half a cup at 3 of the 4 water stations and felt like i needed more with about a mile to go. I don't drink on training runs up to 13M either.
Good Luck 