Please note that Lufthansa checks during boarding whether the carry-on baggage rules are being adhered to. Carry-on baggage that is too large or too heavy will be taken at the gate and loaded into the cargo hold for a fee. The fees for this are significantly higher at the gate than in the check-in hall and can only be settled by credit card.
In my experience Lufthansa group (including Austrian and Swiss) are more zealous about this than other airlines. In particular they have a 8 kg weight limit that they occasionally enforce. Sometimes they weigh everything thing with wheels or that looks large or heavy.
If possible, I may want to also bring a backpack in addition to the suitcase.
That's fine. You are allowed an additional "small item". There is a size restrictions but I never had an issue with a "normal" sized backpack.
Another small item of baggage (max. x=30 cm y=40 cm z=10 cm, e.g. handbag, laptop bag).