Using manual egg beater to cream eggs and sugar
You have a whisk; the object in your picture is a whisk and you can use it to cream ingredients for your recipe. The gearing system makes it easier to get a high speed but you could even use a fork or a (totally manual) balloon whisk once the mixture is soft enough.
I would avoid using an immersion blender as it will be hard to get air into the mixture without careful technique; see this related question for more.