Frank de Boer's side recovered from a goal down in their opening game in the competition against Paris Saint-Germain at the Amsterdam Arena to grab a point but their away form is cause for concern in what is a difficult group.
If the four-time winners are to progress, they will surely have to change their fortunes on their travels, where they have not tasted victory since beating Dinamo Zagreb in December 2011.
But the Dutch champions travel to Cyprus in high spirits after hammering NAC Breda 5-2 away on Saturday - which saw Icelandic striker Kolbeinn Sigthorsson net a hat-trick.
It is the first meeting between the two sides but Ajax have happy memories of facing Cypriot teams. Their last visit to the country ended in a 3-0 win over Apollon Limassol in the 2001/02 edition of the UEFA Cup.
Danish midfielder Lasse Schone missed the win over Breda at the weekend with a groin injury and is a doubt, while De Boer will be without Viktor Fischer who has been sidelined since February with a hamstring problem.
Despite not having any points yet, APOEL will have taken huge encouragement from their performance at Barcelona, where they came away with a credible 1-0 defeat.
Their home form in the competition is mixed, having won three and lost four in their two previous Champions League appearances, but in qualifying for the tournament, they defeated HJK Helsinki and Aalborg without conceding in both games at home.
Giorgos Donis' side won their domestic league last year by racking up an impressive 15 wins, losing once in the process.
And they have carried on that impressive form this season. In their last seven outings in all competitions, APOEL have tasted defeat just once - against Barcelona - and won five of the others.
The other game was the 0-0 draw with AEL Limassol at the weekend.
APOEL have injury concerns over defender Anastasios Papazoglou and striker Rafik Djebbour, who are both doubtful for Ajax's visit.
Donis and De Boer lock horns as managers on Tuesday but are no strangers, having been on opposing sides in the 1996 Champions League semi-finals as the Dutchman's Ajax prevailed against the Greek's Panathinaikos.
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