Naughty streamer even I watch.
TIMER (watch) — anagram ("naughty") of .T.E.M.R (even letters of "streamer") and I
Justify the new mass rite.
MERIT (justify) — anagram ("new") of M (m, "mass" in physics) and RITE.
Not a grey cap on the board?
MITRE — I'm not sure here. A mitre is a cap where two boards join at an angle. It is also a bishop's hat, and a bishop is a man on a chess board? "Not a" suggests that A can be inserted to MITRE to make a word meaning grey. The word is "maître", but I can't see its relation to grey.
Send up Queen's University.
REMIT (send) — reversal ("up") of ER (Elizabeth Regina, queen) + MIT (University in Cambridge, Massachusetts)
Notice a pattern?
These are all anagrams of each other. We are looking for a fifth one. (There's miter, but that's only the American spelling of mitre, so let's discount that.)
Jules Rimet was the FIFA president who introduced the Association Football World Cup tournament. The original World Cup Trophy is called the Jules Rimet Cup in his honour. In 1970, it was awarded permanently to Brazil, the first team to win it three times.