Novel monoterpene-based hydroxamic acids of two structural types were synthesized for the first time. The first type consisted of compounds with a hydroxamate group directly bound to acyclic, monocyclic and bicyclic monoterpene scaffolds. The second type included hydroxamic acids connected with the monoterpene moiety through aliphatic (hexa/heptamethylene) or aromatic linkers. An in vitro analysis of biological activity demonstrated that some of these molecules had powerful HDAC6 inhibitory activity, with the presence of a linker area in the structure of compounds playing a key role. In particular, it was found that hydroxamic acids containing a hexa- and heptamethylene linker and (-)-perill fragment in the Cap group exhibit excellent inhibitory activity against HDAC6 with IC50 in the submicromolar range from 0.56 ± 0.01 µM to 0.74 ± 0.02 µM. The results of the study of antiradical activity demonstrated the presence of moderate ability for some hydroxamic acids to scavenge 2,2-diphenyl-1-picrylhydrazyl (DPPH) and 2ROO• radicals. The correlation coefficient between the DPPH radical scavenging activity and oxygen radical absorbance capacity (ORAC) value was R2 = 0.8400. In addition, compounds with an aromatic linker based on para-substituted cinnamic acids, having a monocyclic para-menthene skeleton as a Cap group, 35a, 38a, 35b and 38b, demonstrated a significant ability to suppress the aggregation of the pathological β-amyloid peptide 1-42. The 35a lead compound with a promising profile of biological activity, discovered in the in vitro experiments, demonstrated neuroprotective effects on in vivo models of Alzheimer’s disease using 5xFAD transgenic mice. Together, the results obtained demonstrate a potential strategy for the use of monoterpene-derived hydroxamic acids for treatment of various aspects of Alzheimer’s disease.