I am not able to figure this out:
Lets say i got an olap-table wihich displays some facts about a product. One of those is the volume of its sales.
How can I add a Field which displays me the Difference to the last year? Provided that I am not allowed to create a new measure, but using a formula?

Something “easy” like my pseudocode:
[Measures].[Volume] From 'FilterYear'
- [Measures].[Volume] From 'FilterYear' -1
Is that even possible? Thank you in advance.
Yes it is totally possible but requires a little bit more MDX trickery than your pseudocode.
Try something like:
This assumes you’re using a year member in some axis; if you selected a day member instead, it would give you volume for the previous day instead.
Here’s a good resource about the Aggregate function and MDX functions in general.