When a new diplomatic officer was assigned to an embassy that was suspected to be involved with espionage, it was standard practice for the counterintelligence agency of the host country to try to discern whether or not they were an intelligence officer. This could be a very lengthy process and would usually include following them, interviewing people they interacted with, etc etc.

Usually within a few weeks the counterintelligence agents would have a good idea of who was an agent and who wasn't. Legitimate diplomats would generally be left alone outside of exceptional cases, but known intelligence agents would be followed and photographed routinely. It was standard practice for all embassies to be watched - the FBI had a 24-hour surveillance operation on the Washington Soviet embassy, which would be stepped up even more if they suspected something was going down.

As for how they entered the building... the normal way? A large embassy would probably have multiple entrances (one for the public, one for staff, one for deliveries, etc etc) but there would not be likely to be a secret entrance of some kind. The best way to enter such an embassy undetected would be to hide in a crowd - a large embassy, such as the Soviet one, would have a constant flow of people during work hours.