Hi ziggy1 .. welcome to Holidaytruths.

You could certainly walk from the new bus station to the Flamingo Benidorm, both lie in the same general area of the resort. I probably would walk if I didn't have luggage and kids. But you may find it a bit of a trek, depending on how hot it is when you arrive and how much luggage you have to carry, etc. If that was likely to be a problem and you didn't want to hire two taxis, I suppose one of the adults could take a taxi along with the kids and luggage.

We have a few other topics which may be of interest and may help you make up your mind:

Alicante Airport Q&As (including resort transfers)
Benidorm Bus Station
Hotel Flamingo Benidorm

David

i booked transfers with travel republic used the shuttle transfer think it was £19 each