Better, one hopes, than Sting (which I don't have).
What's interesting about Dowland's music is how every piece/song can be interpreted differently, whether it's through different instrumentations (lute solo, lute/viols, voice/lute, voice/viols...) or voice selection (counter tenor, soprano, solo voice, groups of voices) and they all can and have been successful.
Then there's a more "modern" or "crossover" approach to Dowland, three examples I can think of are Sting, the "In Darkness let me dwell" disc on ECM and The Forge Players/Freddie Wadling.
The "In Darkness" disc features performers on lute and tenor John Potter, but it also adds a bass clarinet, sop. saxophone and other solo strings. It seems to be an attempt to create another-wordly sound of Dowland, an alternative way of performing his music, allowing improvised lines from the sax and other liberties such as are taken, it feels much more of a Dowland Remixed album, some successful, but some songs and melodies get lost in their interpretations. I find myself enjoying the album, but not necessarily as a Dowland album.
Sting, I feel, failed at his attempt. He simply sang songs accompanied by a lute, nothing more or less, in most cases this has worked, but there's nothing revolutionary in his approach, and he offers very little emotion or style which results in what feels like an hour of the same song. Emma Kirkby or Gerard Lesne create each sung line with emotion.
The Forge Players/Freddie Wadling (and this based on samples) seem to approach the music as written, but merge it with their own individual tones. Almost as if to say, this is not how we think Dowland should be played, but rather this is what we sound like when performing Dowland.