Before their current cycle of five wins followed by five losses, the last time the Mets had a five game losing streak followed immediately by a by a five game win streak was the first 10 games of Willie Randolph's managerial career. The Mets lost his first five games, April 4 to 9, 2005, then immediately won the next five in a row (the Mets also won the next game to stretch the win streak to 6).
The Mets also won five in a row in September of that season, the second win of which, on September 24, 2005 (a Tom Glavine sparkler -- 8 IP, 2 R against the Nats), got the Mets to .500 for the season at 77 wins, 77 losses. The team has not been under .500 at any time since that day. Willie Randolph as of today is 54 games over .500 for his career as a manager -- the 72nd highest total of all time according to http://www.baseball-reference.com/managers/ .
BTW, although the Mets are under .500 against the Phils (6 wins, 9 losses) and Braves (7 wins, 8 losses) this season, the Mets have outcored them both in their head-to-head games. In games against the Phillies the Mets have scored 71 (4.73 per game) and given up 70 (4.67 per game), in games against Atlanta they've scored 65 (4.33 per game) and given up 58 (3.87 per game).