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There are 2 route options from Ankara to Avanos;one is slightly longer but could be quicker as it is on motorway for the most part.The other is shorter but could be a little slower as it goes through some cities and towns.It is still a main road and in good condition but it will be a bit slower.
The bus takes 5 hours and this is leaving from AŞTİ the main otogar in the city centre,you would be coming from the airport which is north east of the city so you wouldn't have to negotiate any heavy city centre traffic,therefore you should be able to do either route comfortably in c.4 hours.
The longer but quicker route is as follows-
from the airport follow signs for the city centre,this will bring you to a major intersection with the ring road E 90.Turn left and follow this south as far as another large junction at GÖLBAŞI where you will see signs for KONYA.This is a continuation of the E90 so turn south here and keep following the Konya signs.From the airport to here will take around 1 hour,maybe a little more depending on traffic.
Further along this road you'll see sgns for ŞEREFLİKOÇHİSAR and AKSARAY.Once in the latter follow the signs for NEVŞEHİR and then to Avanos.

The other route option is-
follow the above directions on to the ring road R90 but leave at an earlier junction with the E88 signed for ELMADAĞ and KIRIKKALE.
Once you reach the latter you follow the ring road around that city and join on to the D765 sign posted for AKPINAR,KIRŞEHİR and HACIBEKTAŞ.This D765 road will lead you to within just as couple of miles of Avanos to a place called GÜLŞEHİR from where signs will take you via smaller roads to Avanos(smaller roads but not cart tracks don't worry!).

Which route you take depends on how confident you feel I guess but both routes are very straightforward to follow and I personally would opt for the latter even though it may be a tad slower.You can also time the journey so you'll know how long to allow for the return leg.

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Just drove both those last month and four hours with one or two quick stops in no problem (unless there are new construction delays) next spring. I too would suggest the second route mentioned.

With an early morning start on April 5 you will have time for some touring. Either follow some of the many brown signs leading to points-of-interest on the way to Ankara or perhaps visit the Museum of Anatolian Civilization in Ankara. Prepare for heavy traffic, confusing streets and big hills on the drive to the museum. I've found GPS systems extremely useful in Turkish cities.