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.

You sure can use it, you will have to adjust your expectations on timeframes though.

Creaming cakes was done long before the existence of electric mixers. I have read (but never been courageous or masochistic enough to try it myself) that, by using a simple whisk moved by one's arm only, it took upwards of one hour of energetic whipping to achieve adequate creaming.

Your whip offers you a mechanical advantage over the simple whisk, so you should need some time between the time needed for manual creaming (1-2 hours) and the time needed by a modern mixer at highest speed (5-10 minutes). I cannot predict where in this interval it will fall, because I've only tried this mechanized type of whisk once, back when I wasn't experienced enough to notice how well it performs in comparison to other methods. Based on my experience with hand-whisking egg whites (with a simple nonmechanized whisk): before you start, ensure you have a backup whisker so you can change every 5-10 minutes, or however long it takes for your arm to get very tired.