If you are on a tight budget, why not have the party at home? At 1, the child is too young to play at a play place by themselves? Another thought is to not "book" a party, but just have one somewhere. Have friends meet at the park where the little one can play and swing, but no expense. As for a cake, I think Sams, Walmart, and Target all do really good cakes for all themes for reasonable prices. However, you can make a cake as well, put a winnie the pooh character on it and call it a day! They sell the candles with the theme characters or do cupcakes and put the little candy things on them with the character. Both should cost less than purchasing a cake.
I understand this is your first child and it's her first birthday, but she won't remember it. All you need are some pictures and good friends to celebrate with. There is NO reason to spend a ton of money!