Hi, nothing really happens or opens until around 10 am, there will be a few eateries open all night but you might well find it intimidating walking in alone in the very early hours as they're not western style places. At around 8 am there will be a few more places open for breakfast and I'd recommend The Baguette just past the palace on the way into town, but there are others, try up Soi 94. You could also go into one of the many hotels and get breakfast but I'm afraid that arriving here between 5-6 am there will be nothing or nowhere to go except walk the streets. I should add you will be very safe to do so providing you keep to the main road area, not that anything is likely if you wander the side streets.
A better option might be to stay at the airport for a few hours and leave there around 6 am, still get you into Hua Hin around 9 with time to eat before going to the water park.