Close synonyms include hop on, mount, mount up, get on, jump on. Each sense on this page lists its own synonyms, because a word with several meanings rarely has one substitute that works for all of them.
Is climb on a common word?
climb on is banded "Uncommon" — turns up in careful writing rather than conversation. The band comes from how often the word appears in a large corpus of everyday English, not from an editor's impression.
Definitions, synonyms and semantic relations for climb on come from Princeton University’s WordNet. Commonness is measured against a corpus of everyday English. Full sources and licences.