unfortunately, no lockers/left luggage service in Nida. Tourist information office does not provide that kind of service either. So you will have to carry your backpack. Maybe you could ask some hotel for that service, but i'm not sure.
Both Vilnius Bus&Train stations have left luggage/lockers. No problems there.
I don't think it's 24 hours. At the bus station I have talked past it late at night when it has been closed so maybe someone knows when it closes?
That should be "walked" past of course........;-) I think it's the same at the train station - closes at some point in the evening.
Luggage storage at the bus station is manned service and operates from 5.25am till 9pm, you can leave your bags for more than a day, more info here, use google translate:
Train station has automatic lockers, they are open 24/7.