At that hour there won't be much of a public transport option (even if the bus was available, it would take you into Port Macquarie town rather than Flynn's Beach).
If you've missed the closing time of the airport desk of your preferred car rental company, taxi from the airport to Flynn's Beach would be the way to go. Whenever I am flying into Port Macquarie to visit the parents, the Qantaslink cabin crew make an announcement asking you to notify them if you want to book a taxi to meet you on arrival - they radio ahead to make the booking. It looks like you're flying Virgin Blue? Not sure if they offer the same service. It's only around 8km trip so the fare won't be prohibitive (taxi fares are government regulated and all cabs are metered),
There's also a limo/shuttle bus operator. Can't comment on them as I've never used them. http://www.limostyle.com.au/airport.html
Budget has an office at the corner of Hollinworth St and Gordon St, Port Macquarie, Phone (02) 6583-5144.
Hertz is at 102 Gordon Street, Port Macquarie, Phone : 02 6583 6599