The current Airservices landing fee for tower services is $10.49 per tonne, which equates to about $17.10 for a BE35. I think Essendon Airport Ltd (the Airport Operators) charge about $23 for their services, which covers parking. Usually there are hard stand tie down positions available, just ask Ground if unfamiliar.
Outside tower hours (after 10pm and before 6am weekdays, 7am Sat, and 8am Sun), there should be no AsA charge because there is no tower service provided. The airspace is still Class C to the ground, administered by Melbourne Approach. It's worth checking ERSA for procedures during this period.
VFR shouldn't attract any other fees.
Getting out of the airport is tricky if you don't have a swipe card. During business hours the Airport Safety Officer can usually assist. Otherwise egress can be gained through a turnstile gate at the northern corner of the terminal at the inner end of the finger. Just follow the instructions adjacent to the turnstile. The terminal building is locked both ways for non card holders between 6pm and 6am I think are the times.
The Operators are about to embark on a complete refurbishment of the airport lighting. All runway and VASIS will be replaced by state of the art LED equipment. To their credit this is a big project for a GA airport.
Hope you enjoy the experience, don't forget to ask for the city orbits.
CG