Following a two-year absence from the Test arena, Biggar made his international comeback against the Wallabies in December.
Now he aims to make the most of his opportunity on Saturday to ensure he makes the trip Down Under for Wales' three-Test tour of Australia.
"To see your name announced is something you want and to come back is a little bit like coming back into school," said Biggar.
"I am just really glad to be back and really looking forward to Saturday and, hopefully, if I get on the tour, pushing for a place out there.
"We are aware that the first-choice team flies out on Thursday and it's up to us, stepping up to the plate and showing we are competitive.
"There are a lot of guys pushing the starters all the way and a good game against the Barbarians will do our cause no harm."
Priestland is one of 16 players to jet Down Under in advance of the Barbarians match to begin preparations for the first Test.
James Hook is the other fly-half contender but lines-up at inside-centre against the Baa-Baas.
Biggar has been in irrepressible form in recent weeks and he demonstrated nerves of steel to slot a late, touchline conversion as the Ospreys snatched the PRO12 title from Leinster, last Sunday.
He has paid tribute to the way his Scarlets rival has made the famous red number 10 jersey his own but is determined to provide some fierce competition.
"Rhys has done superbly well over the last 12 months," added Biggar. "He quite rightly deserves to be number one in the squad at the moment.
"We know where we are in the pecking order and it is about trying to change that pecking order. There is no point me being happy to be on the bench, or having a holiday, because I want to be pushing all the way.
"If I can push him, it will make him a better player and Wales a better team, then we are all winners."