I wonder which year you have to count the farthest in a draft to find a cup winner? None of the top-5 picks in 1999 won a cup. Not sure how common this is, but it was the first year I checked.
In fact, for 1999 I had to count up to the 21st pick to find Nick Boynton, who eventually got his name on the Cup with the '10 Blackhawks. Ouch.
It looks like the next guy in that draft to win a Cup is Mike Commodore at 42 (very quick eye scan tbf). So two Cups total in the first 42 picks of the draft.
Since it's a fun exercise, after a quick glance (feel free to correct me, arbitrarily started at 1995):
1995: Chris Kelly (6th overall, spare forward for the 2000 Devils)
1996: Boyd Devereaux (depth forward on 2002 Wings)
1997: Nick Boynton (9th overall, depth D on 2010 Hawks)
1998: Vinny Lecavalier (1st overall, 2004 Lightning)
1999: Nick Boynton again (21st overall)
2000: Marian Gaborik (3rd overall, 2014 Kings)
2001: Patrick Sharp (95th overall, 2010 Hawks)
2002: Jay Bouwmeester (3rd overall, 2019 Blues)
2003: Marc-Andre Fleury (1st overall, 2009 Penguins)
2004: Alex Ovechkin (1st overall, 2018 Capitals)
2005: Sidney Crosby (1st overall, 2009 Penguins)
2006: Jordan Staal (2nd overall, 2009 Penguins)
2007: Patrick Kane (1st overall, 2010 Hawks)
2008: Steven Stamkos (1st overall, 2020 Lightning)
2009: Victor Hedman (2nd overall, 2020 Lightning)
2010: Tyler Seguin (2nd overall, 2011 Bruins)
2011: Brandon Saad (43rd overall, 2013 Hawks)
2012: Tom Wilson (17th overall, 2018 Capitals)
2013: Andre Burakovsky (23rd overall, 2018 Capitals)
2014: Jakub Vrana (13th overall, 2018 Capitals)
2015: Mitchell Stephens (33rd overall, 2020 Lightning)*, Vince Dunn (51st overall, 2019 Blues)
2016: N/A?
2017: Robert Thomas (20th overall, 2019 Blues)
2018: N/A
2019: N/A