Alonso looked fast throughout the first day of the 2023 season and took the top spot in the late afternoon FP2 session after lapping 0.169s faster than the Red Bull of Max Verstappen.
His team-mate Lance Stroll was sixth fastest, despite the Canadian missing last week's testing due to a wrist injury.
After a strong performance by Alonso in testing, rival teams tipped Aston to be in the mix with Ferrari and Mercedes, but the car was not expected to outpace Red Bull.
Asked if would now be disappointed if he didn't get pole on Saturday, Alonso was keen to keep expectations in check.
"Oh, no, not at all," said the Spaniard, whose last pole position came in the 2012 German Grand Prix.
"I'm not thinking that high, to be honest. Actually, I don't know exactly what position will be a good one for us.
"After testing, we were thinking to be in Q3 with both cars, and score as many points as possible in these first couple of races, try not to make mistakes.
"It's very easy to make a mistake, it's not the same fighting for P12, P14 as fighting for top five positions, because the pressure is different, the adrenaline is different. So there are a lot of things that we as a team, we have to grow together, in this in this process.
"I expect that we will, unfortunately, make some mistakes, I maybe make some mistakes, it's a completely new team, new procedures, the team as well. So we have to be with the feet on the ground.
"The target has to be fighting for the championship, but in the long term, I don't think in this year yet."
Alonso praised the AMR23 but was keen to point out that there was more to come in terms of both improvements in the car and in the way the team operates as he continues to adapt to the Silverstone outfit.
"Obviously, it's good to see the times, that you are competitive," he said.
"After testing there are always mixed feelings where you are, and still as we go into qualifying and the race. And maybe the first three races are very different circuits, Bahrain, Jeddah and Australia.
"At the moment we are just concentrating on ourselves. Still a lot of things to improve in the car, the balance was not completely perfect today.
"And I think also the team has to change a few things, we are in the process of trying to reinforce every area of the team."
Alonso also hailed the efforts of the team in recent months to move forward. Aston finished seventh in the constructors' championship last season, beaten to sixth on countback by Alfa Romeo.
"Obviously, it feels good to drive, I will lie if I say that it's not feeling good," he said.
"But everything is relative to what the other teams are doing. And you can feel very good in one car, and maybe there are three or four cars faster than you, and then it doesn't feel that good anymore.
"So let's see where we are. I think we have to be proud, the step that we have done, it has been a very intense, two or three months' work in the factory.
"We spend endless hours in the simulator, in the meetings, and try to anticipate what problems we could face in these first couple of races.
"And as I said the knowledge that this team has, the talent that this team has, is probably unprecedented. The group of people we have. So that's something that is very encouraging for the future."
