Is the Hydrogen spectrum only an emission spectrum?
Emission and absorption are inverse processes.
So, naturally, if an atom exhibits some emission spectrum, it will also give an absorption spectrum. So you were right in assuming that.
However, the processes by which these two spectra are generated are completely the opposite of each other.
An atom emits a photon when an electron goes from a higher energy state to a lower energy state, thereby releasing energy in the form of light. When a photon is absorbed, the electron uses this energy to excite to a higher energy state.
What kind of spectrum you have at hand depends on which instrument and detector you are using. When atoms releases photons (light), the detector catches the production of the photon and gives out emission spectra. For absorption spectroscopy, varying wavelengths of photons are incident on the atom and the corresponding photon is absorbed by the atom resulting in no photon of that particular wavelength being detected.
I am attaching the images of the emission and absorption spectra of hydrogen atom. Note how they are complete inverses of each other.

Image source: https://chem.libretexts.org/Courses/Solano_Community_College/Chem_160/Chapter_07%3A_Atomic_Structure_and_Periodicity/7.03_The_Atomic_Spectrum_of_Hydrogen