Old Town is easily reachable by bus or tram in about ten minutes from the main station and the Petrzalka station. At the bus stop or tram stop, look for destinations “Novy Most” or “Kapucinska”, the closest to most Old Town destinations  or walk 15 min from station to President’s Castle down Stefanikova street. 1.5 hours would be enough time to do the walk.  Go to Presidenck Palace (President’s Castle have a little look around there, walk up Michalská, through the Michael's Gate gateway and then you can walk up to Hodžovo námestie- major square in town , after that visit St.Michael’s Tower, proceed to Stara Radnica (Old Town Hall) , go over to Primate’s Palace , then walk along to St.Martin’s Cathedral and if you still have time go further on uphill to Bratislava Castle. Take a train or bus back to station or walk from the Bratislava castle for 25 min. 

Don’t forget to eat famous creamcakes!