The Newcastle youngsters were invited to meet the Football Association's president following their exploits at the FIFA Under-20 World Cup in South Korea.
Armstrong scored during the competition while Woodman saved a penalty in the final as the Young Lions - led by former United coach Paul Simpson - lifted the trophy with victory over Venezuela.
Woodman also won the Golden Glove award for the tournament's outstanding goalkeeper.
The Newcastle pair were among 16 players who joined coaches and staff in meeting the Duke, fresh from taking his son Prince George to his first day at school that morning, at the Palace.
Armstrong is currently on loan with Sky Bet Championship side Bolton while Woodman returned for training with Newcastle on Friday, having played for his country at under-21 level earlier in the week.
Speaking to the official FA website, coach Simpson said: "It's been brilliant for us to be able to see this and for the Royal Family to honour the players, it's a lovely gesture.
"We just had a general chat with HRH and he was saying how proud he was and how we've set the benchmark, not only with the men's development teams but also with the women's teams and the ethos that they've created there.
"He's really proud and is hoping for these groups of players to progress into the senior team and get success down the line, which is what we all want.
"I've never seen the lads behave as well as they have done! They all spoke very well to the Duke and it's been a fantastic day."
Watch an exclusive NUFC TV interview with Freddie Woodman here.
Photograph courtesy of the Football Association