Put simply, no.

The Jewish Quarter was largely destroyed, not just denied as far as access goes. The armistice line was sealed, and the Old City was entirely blocked off from any Israeli entry. Exceptions were few and far between, typically made for Arab Christian entrants seeking to enter Jordan generally, or cases of seeking to visit a dying family member, or something along those lines.

There was no Israeli Jewish access to the Christian and Muslim quarters of the Old City. Israeli Arabs could sometimes gain access in extremely rare cases with good cause, but both countries criminalized attempts to cross without authorization, and authorization was rarely given. Israeli Arab Muslims were granted access less often than Christians by some accounts, due to a desire perhaps to maintain good public relations with the Christian world. Israeli Arab Christian entrants were ostensibly there to visit holy sites but often went to see family for as long as they could before returning.

Jordanians rarely entered Israel either, though some were admitted for medical treatment not available in Jordan.