You could get a rental car. For example Sixt will bring it to where you are in Reykjavik (or you can get it at the airport), and you can drop it off at the airport. I presume other companies do this too. Don't know about the times though... what time they can deliver it to you. Perhaps try and get it the evening before you drive.
You don't need a 4wd, a normal car will do if you stick to the Golden circle or part of it, for example Geysir (with Strokkur geyser going off about every ten minutes!), and Gulfoss (very beautiful waterfall). These are open sites with no charge.
A dip in the Blue Lagoon is also lovely (although expensive), I think there are buses between that and the airport, and also to / from Reykjavik.
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