At minimum, the Destiny's shields are not better than Atlantis's shields.
Destiny is the oldest1 Ancient technology we have ever seen on-screen. It was launched while they still lived on Earth, before they built the Milky Way's gate system - its gate technology was far more limited, and only able to reach a couple Stargates in the galaxy at a time. Likewise, the gate address to Destiny was stored in the database using the Milk Way gate system's glyphs.
In Stargate: Atlantis, the Daedalus's Asgard shields were shown many times to vastly more powerful when powered by an Ancient ZPM. However, it has never managed to stand up to multiple Wraith hive ships for very long. Atlantis's shields on the other hand, can last an extended amount of time against multiple hive ships.
As far as I know, there has never been a direct comparison that would let us determine how strong Asgard shields are compared to the Destiny. Atlantis's shields may or may not be stronger than Destiny's shields, but they're definitely not weaker. However, I would bet that they are stronger due to the huge time span between when Destiny was built and Atlantis was built.
1 Well, just about the oldest. There are a few scenes in the movie The Ark of Truth that are from even further in the past, showing some Ancients escaping the Ori. In the second one, we see some pages in a notebook that show a Stargate being designed, with its owner saying "I had the most amazing idea last night...". Some of the technology in the background is reminiscent of Destiny's technology, or what it might have looked like before it had been run down by age.