I want to take my bike as a luggage from Bratislava to Vienna. Do you know if any regular bus line (specifically those departing from bus station at Mlynske Nivy) will accept bike?
If not, can I take it with train?
You can take your bike on a train to Vienna. From ÖBB website (the national railway system of Austria):
www.wien-vienna.at offers transport by boat between Vienna and Bratislava. One way ticket costs 23 EUR and bike transport costs 4 EUR extra.
There is a train Bratislava Main St. - Vienna Main St. every hour. Bikes can be taken without extra charge. (Adult fare was 13€).