Renting a car to "Get around" within Athens itself is a bad idea, very bad. To make a comparison, if you were to be in Manhattan (NY) for a weekend, staying on say, Park avenue & 56, would you rent a car to get around Manhattan??? Of course not! traffic traffic traffic and NO place to park, $$$$ to garage, and plenty of public transport a few steps away. Athens even more so, because landmarks are almost all in center, within walking distance, and many are on no-cars lanes.
You really don't have "a day in Athens" anyway, because by the time you collect bags, get into the city & check into your hotel, it will be noon. Likewise, when you return on Thursday, it probably will be 6 pm by the time you're in your hotel. So relax, and just stroll around enjoying views of places from the outside. If you're in a Museum Mood, on Thursdays, Benaki Museum is (1) FREE and (2) Open til Midnight!! A super museum! not gigantic, very "do-able" and so well signaged (in English) that even w.o. background in Greek history & art, you can appreciate it. What I like is it gives top-quality "samples." I.E., not a roomful of red vases, just one case of really nice red vases! PLUS, it has a super upper-floor restaurant where savvy chic Athenians meet to eat, drink & "be seen."
This useful link http://www.athensinfoguide.com/open.htm lists all ancient sites, landmarks, museums etc, w. opening hours etc. All museums & ancient sites closed Easter Sunday, but some can be enjoyed as "walk-by"... ie, Hadrian's arch, and, thru the railings, Temple of Zeus. If you walk from Temple of Zeus west along the lovely pedestrian walkway south of the Acropolis, you can look up for BEST view of the Parthenon, to photo yourselves against. Another place just E from Zeus across National Park is the Panathinaiko Stadium (this is the stadium, on actual site of Ancient Games arena, which was built for modern revival of Olympic Games, in 1896). This online Map - http://www.orangesmile.com/destinations/img/athens-map-big.jpgis -- is handy to orient yourself to central Athens. Printable smaller map: http://www.planetware.com/tourist-attractions-/athens-gr-ath-ath.htm#GR107 For both, click to enlarge. If u have time (between picking tablecloths and wedding-party garb), read up a little on Athens, it will increase your pleasure.
For your first night -- Easter -- alas, many many restaurants/tavernas will be closed, even tho they could benefit from tourism, because if they're family-run, often they themselves are celebrating. (Similarly to Xmas Day in US). You might inquire ahead to your Athens hotel, for Easter-evening dining recommendations of NON-hotel places to dine (of course a big hotel will tout its own restaurants otherwise) ... when you return, if you want to "splash out" for your last night in Greece, you could ask (on your FIRST night, the 12th) for your hotel to reserve you a seat for THursday 4/16 at STROFI http://www.strofi.gr/en/ -- read Trip Advisor reviews carefully in advance & choose menu items accordingly.
Congratulations on nuptials and on your FIRST trip to Greece (warning; it can be addictive!).