Apart from weekend visits, the family has been separated since then, with Mr Matthews, a 23-year-old spray painter, remaining in Masterton in order to provide for the family.
Having started a new job in February, he has little annual leave and had already used some of it when Noa was born.
Mr Matthews contacted Work and Income and IRD for help but was told limited assistance was available as his partner's maternity leave was due to kick in next month, he said.
Being so far away from his family was hard, Mr Matthews said.
"It's horrible knowing that if something goes wrong I just can't be there. Even if I jumped in my car right now, if something was going seriously wrong, it would still take me over an hour to be there."
Noa will remain at the neonatal intensive care unit until she weighs enough to undergo a heart operation at Auckland's Starship Hospital, while her mother, Danielle Bolt, stays at Ronald McDonald House.
The family has set up a Givealittle fundraiser to help enable Mr Matthews to take unpaid time off work to be with his family.
So far the page has garnered about $1900 in donations in just over 10 days.
Noa's heart problem came as a surprise to the couple, who had been having monthly scans, Mr Matthews said.
"50 per cent of babies with Down Syndrome are born with holes in their heart, so they paid very close attention . . . they assured us that there was nothing wrong and her heart was fine.
"So we thought we had nothing to worry about, so it was a huge shock."
"Once she was born we were celebrating - we were like 'oh my gosh she's here, she has a healthy heart, everything is going to be fine', so it was pretty heart-wrenching."
He never regretted the decision to have Noa, Mr Matthews said.
"Every minute that I spend with her is just joy. She's amazing. She's just been such a wonderful baby - she doesn't cry about anything."
"There was lots of outside pressure from a lot of people we knew due to lack of information and education - saying we should probably terminate, you guys are too young ... but we did lots of research and watched lots of videos and ended up becoming really excited about it."
"It was all about a lack of education - once we found out it what Down Syndrome actually was, it wasn't such a scary thing."
Anyone interested in donating money to help Mr Matthews take unpaid leave to be with Noa can visit the Givealittle page