City have regularly tangled with European heavyweights in the round-robin stage since their competition debut in 2011-12, but they will be favourites to progress from a Group F containing Shakhtar Donetsk, Napoli and Feyenoord.
Brazil international Fernandinho is set for a reunion with Shakhtar, who he left to join City in 2013, while Pep Guardiola pitting his coaching wits against Napoli's Maurizio Sarri will make for two mouth-watering clashes.
Begiristain believes the group is not without its pitfalls but concedes things could have been worse.
"Of course, I think it is very difficult to go [to face Shakhtar] as they are always building good team. They sign good players from South America," the former Barcelona director told BT Sport.
"In the rest of the group, Napoli now are, for me, the best footballing team in Italy. We saw that against Nice in the play-off round so we have to be careful.
"Also, Feyenoord play good football so I don't know if I am happy or not.
"There are worse draws. If I was Tottenham I would be more worried."
City have only progressed beyond the round of 16 once, when they fell to Madrid at the semi-final stage in 2015-16.
Begiristain has overseen in excess of £200 million investment into Guardiola's first-team squad during the current transfer window and insists a more consistent European return must be the goal at the Etihad Stadium.
"We are an ambitious team. We try to beat the good teams. We are improving our squad and the target is to get as far as we can," he added.
"We want to be a football club who are always there in the quarter-finals and semi-finals, getting to the final.
"We want to be consistent in our performances. I feel we are ready to do it. We have refreshed the team
"I hope we can remember what we did two years ago, going to the semi-finals and why not even more?"