The Eagles sit bottom of the table following a woeful start to the campaign in which they failed to record a single point or even score a goal from their first five matches.
Awaiting Roy Hodgson's charges in the next four fixtures are the division's current top four - Manchester City, Manchester United, Chelsea and Newcastle United, with only the Blues game at Selhurst Park.
Despite the seemingly daunting task, Palace captain Puncheon is feeling optimistic, given that his side won away at Chelsea and Liverpool last season, either side of beating Arsenal at home.
Ahead of Saturday's trip to the Etihad Stadium, Puncheon told a pre-match press conference: "We like playing against the top teams — I don't know why.
"But the Premier League is always about three points no matter who you are playing and we'll just keep our heads up. We go into every game trying to win it and I don't think we will ever set our sights not to.
"People say, 'They are the top teams' so obviously the odds are in the top teams' favour. But we are more than strong enough characters to deal with it. This group of players, it will never be for a lack of effort. We will always keep trying, we will always go down fighting."
Despite their Premier League troubles, Palace managed to secure a 1-0 win over Huddersfield Town in the EFL Cup on Tuesday night.