Magnetization and ultrasound measurements have been performed in pulsed magnetic fields up to 60 T on a ferrimagnetic HoFe₅Al₇ single crystal (Curie temperature T_C = 216 K, compensation point T_comp = 65 K) with a tetragonal crystal structure of the ThMn ₁₂-type. The compound exhibits a high magnetic anisotropy of the easy-plane type. A large anisotropy is also observed within the basal plane having an easy-magnetization direction along the [110] axis with the spontaneous magnetic moment M_s = 2μB/f.u. at T = 2 K. Along the easy axis, two first-order field-induced magnetic transitions are observed. At both transitions sharp anomalies in the acoustic properties are found. The critical fields of the transitions depend on temperature in a different manner. Within molecular-field theory and using the high-field magnetization data the Ho-Fe inter-sublattice exchange parameter has been determined to be n_HoFe ≈ 4 T/μB. The magnetoelasticity has also been probed by magnetization measurements under hydrostatic pressure. T_C decreases with a rate dT_C/dp = -10 K/GPa, whereas T_comp increases with dT _comp/dp = 3.5 K/GPa.