Both flights will depart from terminal A.
Regarding Priority Pass: The Hugo Junkers Lounge is located in terminal B, which is connected by walkway from / to terminal A. But you will have to clear immigration and security and pick up your luggage if SQ in SIN is not able to route them to your final destinations LIS. As far as I know shower facilities are available. The Sheraton Hotel - connected to the terminal - also offers rooms for day use (just in case).
My recommendation would be to try to check your luggage to LIS before heading into town. This should be possible as the counters of Eurowings are usually staffed the whole day and DUS could "park" the luggage till departure.
Luggage storage is available at DUS on the ground floor or parking garage 3 (
https://www.dus.com/en/services/gepaeck/left-luggage) for 4,50 EUR / day / piece of luggage up to 30kg.
There are plenty of options to travel into downtown Düsseldorf.
The most easiest would be take the S-Bahn (S 11) from Düsseldorf Airport Terminal A/B/C to Düsseldorf main station (Hauptbahnhof) and from there any subway (e.g. U 76, U 78, U 79) in direction Heinrich-Heine-Allee. There you are in the heart of downtown Düsseldorf and be able to walk to either Königsallee (for window shopping) or the river Rhine. Depending on the weather you could walk toward the area called "Medienhafen" with modern buildings.
I am not so much a church goer myself, so cannot comment on this. But I would recommend not to travel to Cologne and visit the Dom on a Sunday.
Information about public transportation is available on
http://www.rheinbahn.com/Seiten/default.aspx or
http://www.vrr.de/en/index.html