The Shakers managed to claim just their second clean sheet of the season, but they also drew a blank at the other end so are now six games without a win.
The point was enough to edge Bury up to 18th place though, while Newport are ninth after losing once in their last eight league games.
"We're all disappointed," said Blackwell. "You can't get better chances than we had in the first half.
"We played some great stuff first half, but if you don't take your chances, you can't win games.
"Marco Navas would have finished his chance in training, but it's the pressure of playing on the big stage.
"I don't want the players panicking. There was some frustration again from the crowd and I think it's unjust.
"Second half Newport turned it into a battle, they brought on some big lads, but we had two or three good chances second half too, and Tom Soares should have scored at least one.
"But we got a clean sheet and dominated the game without finishing it off so I can't complain too much because we did well. If we can just finish off the approach play we'd be ok."
Navas, brother of Manchester City winger Jesus, was handed his first Bury start and sliced an early shot wide.
Anton Forrester's cross then found him unmarked, but the Spanish winger somehow headed wide.
Newport threatened with Danny Crow and Lee Minshull going close, while Adam Chapman shot straight at Bury keeper Brian Jensen after the break.
Forrester headed over while Soares had one shot saved by Lenny Pidgeley and sent two off target.
Yet Newport went closest to a late winner through Minshull and Chapman.