Since the iPhone 4 is not the earliest model available, the iOS system itself and the apps you use with it can be somewhat slower and in some cases even unusable (those you shouldn't be able to download from the app store). The Spotify app though should work on your iPhone 4. If the app crashes on startup this is most likely caused by a lack of enough RAM memory. In human language, there are to many applications using memory preventing Spotify to reserve enough space for itself.
You can try to double tap the home button and closing all your applications (including spotify) and then trying to launch Spotify again. This should solve the RAM issue.
If that doesn't work re-installing Spotify is also a possibility but you might want to start with a less drastic measure;-)