The young Magpies, currently occupying fourth spot in Division Two, travel to the Riverside Stadium to face their North east neighbours in their first of three successive away fixtures (kick-off 7pm BST).
Brookwell, who has captained United's under-23 side in their previous four league matches, is determined to end the 2021/22 campaign on a positive note.
"We're all very determined to finish the season strongly," the Wigan-born 20-year-old told nufc.co.uk. "We don't want to waste everything we've done in the season and want something to show from it, whether that is winning the (Northumberland) Senior Cup or getting into the play-offs and gaining promotion. We don't want it to dwindle out.
"It's always good when you play at the stadiums and it is a bit of a derby against Middlesbrough. It's also a game we want to win to help us try and get promoted so it's a big game for a few reasons.
"Everyone wants to get to play at big stadiums every week and it gives you a taste of what it's like. There might not be as many fans there but it's still good experience to be playing in the grounds.
"Whether we're home or away, we believe that we'll play the same way so it doesn't really matter where we play. We're looking forward to the three away games and seeing what we can get out of them."
Newcastle's youngsters were involved in a seven-goal thriller against West Bromwich Albion on Monday night, recovering from a two-goal deficit at half-time to seal a 4-3 dramatic victory at St. James' Park, with Josh Scott scoring a stoppage-time winner for the hosts.
Brookwell was on target against the Baggies, netting United's second goal as well as recording an assist, and admitted he was not predicting such a significant impact in front of goal after featuring at the heart of Newcastle's backline.
"I probably don't think I was expecting to score and assist playing as a centre-back but I'll definitely take it," the versatile defender, who joined the Magpies' Academy from Liverpool in the summer of 2020, added.
"The lads were happy with the win. We thought we made it harder for ourselves than it should have been but, obviously, we weren't bothered by that in the end once we'd won.
"It's always better to win games like that - you'd rather win like that than any other way. It was good to score and to say I've scored at St. James' Park is a good thing to be able to say. That was for the equaliser and we all knew the goals were going to come.
"Luckily, they came early enough in the second-half for us to kick on and give ourselves enough time to grab a winner.
"I probably don't think I was expecting to score and assist playing as a centre-back but I'll definitely take it."