We present a first-principles study of the electronic transport properties of metallic and semiconducting carbon nanotube (CNT) junctions connecting two graphene layers, for different CNT lengths and link structures. Transport is analyzed in terms of the scattering states originated from the π and π* states of the finite-length CNTs, which couple to the graphene states producing resonances in the transmission curves. We find that, for metallic CNTs, the conductance is nearly independent of the tube length, but changes strongly with the link structure, while the opposite occurs for semiconducting CNTs, where the conductance in the tunneling regime is mainly controlled by the tube length and independent of the link structure. The sizable band offset between graphene and the CNTs yields to considerable effects on the transport properties, which cannot be captured using simple empirical models and highlights the need for a first-principles description.