Barrett arrived in the Irish rugby stronghold last week and has been training with the Leinster side and watched them play Ulster last weekend.
Ireland prop Tadhg Furlong picked him up upon arrival at Dublin Airport.
"It's nice to be able to rub shoulders with a lot of boys this week, the internationals are away, so I've been seeing plenty of fresh face in and around the environment, and it's a special time that I'm very much looking forward to."
Since signing up, he told Leinster Rugby TV that he has been looking forward to his stint at the club.
Last month, he got a taste of what to expect in Dublin with the All Blacks for their Test match.
"It was a funny feeling playing against a few of the Irish boys who are here with Leinster and knowing that I'll be hopefully chucking on a jersey beside them in a few weeks time, so it's pretty special."
He will have Kiwi support at the club as former Hurricanes back coach and Crusaders player Tyler Bleyendaal is an assistant coach.
"He's an outstanding coach. He's a guy I have leant on a lot in the past few years at the Hurricanes. He's well respected there, and I'm sure he is also here at Leinster.
"It's probably as good as I could have dreamed for in terms of continuity, my game, and trying to grow as a footie player.
"I'm looking forward to the growth and the relationship that we can continue here. He is a great guy too, so he's going to add a lot of value for Leinster."
Barrett said he had been watching the club's performances this season.
"Leinster have started very well, used a number of players and have plenty of depth. They were massive against Munster. I was blown away. Sitting at home in New Zealand, it was hard to fathom seeing 80-odd thousand at Croke Park. Unbelievable.
"It would be amazing to play in front of the loyal Leinster fans and even people throughout Ireland.
"Normally, this time of the year, I'm on a plane heading home to enjoy some time off in the summer, but my body's feeling great.
"I'm excited and can't wait to get stuck into the URC and some European Championship games, hopefully. It can't come quick enough."
Meanwhile, another Wellingtonian, former Super Rugby-winning coach Chris Boyd has joined Munster as a performance consultant.
Boyd, who coached Northamptonshire in the Premiership and European rugby, has been employed to support the wider Munster coaching group over the remainder of the season.