I've just inherited a small amount of cash, which hopefully I'm going to
use to visit Oz and NZ this year.

Am I right in thinking I can get a visa waiver as I'm from the UK?

You can get a Visa "online" from the Aus Government site

http://www.eta.immi.gov.au/

Dont get "tricked" by other, more costly "private-for profit" websites

Do I need a place organised for my first night in Oz, as you do when you
fly to the US?

Its my understanding that nearly ALL the flights to the major Ports of Entry
in AUS arrive in the EARLY AM having flown "overnight"
...such were all the options that I checked into