If you enter Germany from a non-EU country, you have to apply for a visa. Make sure you apply for this visa in good time at the German diplomatic mission in your home country.
EU nationals and nationals of certain countries (e.g., Australia, Great Britain, Iceland, Israel, Japan, Canada, Liechtenstein, South Korea, New Zealand, Norway, Switzerland, and the United States) do not need a visa to enter Germany.
More information: Entry and residence regulations
Registration at TUHH takes place under the status of "guest researcher," not under the status of "exchange student." As a guest researcher, you will have access to the infrastructure of the TUHH campus.
Please contact the STUDIS department, which will assist you with this. The responsible staff member, Ms. Hennke, can be reached at Vanessa.hennke(at)tuhh(dot)de.