Se você pretende fazer isso para cada linha, tente isto:
SELECT a1.id,
(SELECT MIN(a2.Auction_StartDate)
FROM Auctions a2
WHERE a2.Auction_StartDate > a1.Auction_StartDate) AS nextStartDate
FROM Auctions a1
SELECT a1.id,
(SELECT MIN(a2.Auction_StartDate)
FROM Auctions a2
WHERE a2.Auction_StartDate > a1.Auction_StartDate) AS nextStartDate
FROM Auctions a1