Gordon Strachan is expected to be unveiled as the new Boro boss on Monday following the sacking of Gareth Southgate late on Tuesday night.
Cooper underlined his support for the next incumbent and said: "I have spent two thirds of my life at this club. I don't want to leave, but that's not my decision.
"I want to work with anyone who comes in. I will give it all I've got. Just because Gareth is a mate doesn't mean I won't support the next guy."
Boro will have been disappointed not to have taken all three points with only Billy Jones' injury-time equaliser denying them.
Gary O'Neil had given Boro the lead just before halftime and although Paul Parry levelled parity lasted just six minutes.
Adam Johnson then restored the lead and it appeared Boro were on course for three points when Jones struck.
Cooper admitted he was disappointed at having to leave with just a point.
He said: "I tell the lads unfortunately it comes a bit too much of a coincidence when you concede two to three late goals. We should be sitting fairly close to the top of the table with the points that have gone begging.
"Hopefully we will crack on and learn and not make mistakes in a game that we should have won comfortably.
"Really, all I am doing is filling a hole. If I am here taking charge of the team today what happens next week happens.
"Even last night I was speaking to Gareth (Southgate) on the phone late on. He wished the team and the club all the best for the future. That's the type of guy he is.
"I was given an opportunity by Gareth to join the first team as a coach two-and-a-half years ago. Until I am told differently I will carry on.
"We are in no worse a position than since we came here."
O'Neil had a 30-yard free-kick tipped over by Preston keeper Andy Lonergan before the Boro midfielder struck two minutes before half-time when his shot thundered through the defensive wall and into the net.
Preston drew level on the hour in a move initially started with Ross Wallace's floated cross. Jon Parkin's attempt was blocked by the keeper until Parry stuck it away at the right post.
Johnson showed his worth by flying down the left flank and planting a tight-angled shot, driven hard, that found its way into goal.
But in the 92nd minute Preston came back for the second time and drew level as Jones headed in at the right post.
Cooper had special words of praise for Johnson whom he admitted was "an outstanding talent".
He said: "Unfortunately there are an awful lot of suitors for him.
"He has already scored eight goals. There is no doubt the boy is going places, that's for sure."
Preston manager Alan Irvine saw his team slip out of the top six, also watched by former North End boss David Moyes.
Irvine said of the result: "I'm not over-happy. We have got a point, but I look for more than just a result. There are a lot of things we will need to take out of that game and improve upon.
"We were poor in the first half. I was praying for half-time to come to address some of the issues that needed addressing.
"I thought we got the response almost from the kick-off in the second half. We then started to stretch the game out a bit better than we had in the first half.
"The lads showed great character again. They don't play as well as I would like at different times, but they keep going and going.
"They deserve enormous credit for keeping going. It's a fantastic quality to have.
"One of the things we identified in our video work was that Boro are a fantastic counter-attacking team.
"We were aware that at any time we were going forward in numbers we had to be aware there might be a counter-attack.
"To get done for the second goal with a counter-attack was extremely disappointing."