Restore new iPhone with only some backup?

My old Iphone 4 which was 32GB was stolen and so I bought a new iPhone 4 but its only 4GB. When I try to restore it with my old iphone's backup it says its incompatible and I'm assuming this is because there's too much memory needed for the smaller space. Is there a way I can only put my contacts on to save memory?

Well first off the iPhone was never available in 4gb. You probably mean the 8gb iPhone. In that case when you update the software to 5.1.1 than you may use iCloud to sync your contacts, calendars, email, find my iPhone, and more. With iCloud all the information is online and on the internet so it doesn't take up much memory.