from Hamburg airport:

• use a Taxi (ca. 25min, about EUR 20,-)
• or take suburban railway S1 (Get ticket from the HVV-ticket machine: Button 3, EUR 2.70) until stop Ohlsdorf,
  change here to metro U1 into direction Ohlstedt/Großhansdorf until stop Kellinghusenstraße,
  change here to metro U3 into direction Billstedt/Mümmelmannsberg (departure is vis-a-vis on the same track) until stop Schlump.
  Exit the metro here, dive up to the daylight, look around for the GEOMATIKUM (an ugly, gray, 18 floors high building), and walk towards it; the little (4 floors), green-blue building nearby on the same side of the road (Bundesstrasse) hosts the Institute of Oceanography (no. 53).
• or take a look at the information webpages of the Hamburger Verkehrsverbund (HVV)

from Hamburg main railway station (Get ticket from the busdriver or from the HVV-ticket machine: Button 2, EUR 1.65):

• take bus no. 4 heading to Eidelstedt to stop Schlump/Bundesstraße
==> now you're within walking distance, walk towards the little skyscraper (Geomatikum), turn left into the Bundesstraße - the 4-floor building to the left of the little skyscraper (Geomatikum) at the right hand side hosts the Institute of Oceanography (no. 53).
• ==> now you're within walking distance, from heading direction turn left into Beim Schlump until you arrive at the Bundesstraße, turn left - the 4-floor building to the left of the little skyscraper (Geomatikum) at the right hand side hosts the Institute of Oceanography (no. 53).
• take U-Bahn (Metro) U2 heading to Niendorf Nord to stop Schlump
  ==> now you're within walking distance, from heading direction turn right into Beim Schlump until you arrive at the Bundesstraße, turn left - the 4-floor building to the left of the little skyscraper (Geomatikum) at the right hand side hosts the Institute of Oceanography (no. 53).

from railway station Hamburg-Dammtor (ICE-Station) (Get ticket from the busdriver or from the HVV-ticket machine: Button 2, EUR 1.65): 

• take bus no. 4 heading to Eidelstedt to stop Schlump/Bundesstraße
==> now you're within walking distance, walk towards the little skyscraper (Geomatikum), turn left into the Bundesstraße - the 4-floor building to the left of the little skyscraper (Geomatikum) at the right hand side hosts the Institute of Oceanography (no. 53).
• ==> now you're within walking distance, from heading direction turn left into Beim Schlump until you arrive at the Bundesstraße, turn left - the 4-floor building to the left of the little skyscraper (Geomatikum) at the right hand side hosts the Institute of Oceanography (no. 53).
• Just walk (ca. 15min): leave the railway station and turn left, walk the Edmund-Siemers Allee to northwest - the railway track is to the left. You'll automatically arrive at the Bundesstraße which you have to follow for about 10min. The Institute of Oceanography (no. 53) is on the left hand side.
• or take a look at the information webpages of the Hamburger Verkehrsverbund (HVV)