Koller's squad are taking part in only their second European Championship finals and it is the first time they have played their way through the qualifiers, having won a place automatically as co-hosts eight years ago.
Even then, they were knocked out in the group stage without winning a game and, when the coach took over in 2011, Austrian football was at a low ebb and considered something of a lost cause in a nation obsessed with Alpine skiing.
Expectations though have rocketed after they steamed through qualifying with seven wins and a draw. Public opinion now views them as dark horses while critics have labelled them "the new Belgium".
"It's very important for the players to be happy and to enjoy themselves," Koller told reporters ahead of Tuesday's opening Group F game against Hungary. "There are very high expectations in Austria.
"We are very aware of what is at stake and, of course, the players are very excited the tournament is finally starting. Our task is to not go overboard and build up to the game in a very clear frame of mind.
"It comes down to one's state of mind," added Koller. "That can take a team very far, it's very important. It's your head that says you will keep going even when you are tired.
"We have had a very good preparation period and it's time to get started. There will be a touch of nerves at the start and I hope those quickly fall by the wayside."
Captain Christian Fuchs said his experience in winning the English Premier League with rank outsiders Leicester City this season had given him added inspiration.
"Now I start to dream and I believe that things can happen but it's not easy, there are good teams in the way," explained Fuchs.
"I've been sleeping very well indeed. Clearly we are very excited things are going to get started.
"You start to think about it a bit more on the day before the first game but we have enough players used to these high-pressure conditions," said Fuchs.
(Writing by Brian Homewood; Editing by Tony Jimenez)
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