The winners of this two-legged tie, with the return fixture to take place in Cyprus the following week, will face Zorya Luhansk in the final qualifying round.
Rapid would have had their sights set on qualifying for the Champions League, but their hopes of doing so were ended by Sparta Prague in the second qualifying round.
The Green-Whites took a 2-1 lead into the second leg thanks to Christoph Knasmullner's double, but they succumbed to a 2-0 loss in the Czech capital to crash out 3-2 on aggregate.
Dietmar Kuhbauer's men have a week to get over that disappointment ahead of the visit of Famagusta, who are entering the competition at this stage after winning the Cypriot Cup.
The Blue and White finished second in the Cypriot top flight last season and are still a couple of weeks away from beginning their domestic 2021-22 campaign.
Famagusta have won all three friendlies they have played during pre-season, though a lack of competitive football may well work against them on Thursday evening.
Rapid have not only played some European ties in recent weeks, they have also played a couple of games in the Austrian Bundesliga.
However, results have not exactly gone to plan so far, having lost 2-0 to Hartberg on the opening weekend before drawing 1-1 at LASK last time out.
The pressure is now on Kuhbauer to get things right, then, but only one of these sides can advance through to UEFA's secondary club competition.
Knasmullner may be tempted to make some changes to his starting lineup on the back of the 1-1 draw with LASK at the weekend.
As for the visitors, Temur Ketsbaia has had a few games to work out his favoured formation and lineup for this contest.
Last week's 4-1 win against PAEEK may offer a good indication of what to expect from Famagusta.
Having lost a few key men over the summer, the worry for Ketsbaia is that his side may take some time to click into gear.
Rapid Vienna possible starting lineup:
Strebinger; Stojkovic, Greiml, Wimmer; Schick, Schuster, Petrovic, Ullmann; Knasmullner, Fountas; Kara
Anorthosis Famagusta possible starting lineup:
Tsur; Hambardzumyan, Correa, Arajuuri, Correia; Husbauer, Artymatas, Christodoulopoulos; Deletic, Roushias, Christofi
We say: Rapid Vienna 1-1 Anorthosis Famagusta
Rapid Vienna have an advantage given that they have already played a handful of competitive games this season, though results have not exactly been great.
Indeed, the hosts enter this match winless in three - including defeat in their Champions League qualifier - and we can see Famagusta taking a draw back to Cyprus with them.
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