As sunrise and length of day will depend entirely on your latitude and longitude I fear that you will have to calculate these locally. You would need a reference table and GPS location as well as the current time (which would be available from a GPS receiver) in order to calculate the values that you need. This is all straightforward mathematically but is not a simple project. So I think the answer to your question would be a GPS receiver and a set of local reference tables. Could this be achieved with a PICAXE? Yes, I think it could.
I have written code to make a crude clock on an M2-series PICAXE. It predicts sunrise and sunset each day so that it can switch a (large) lighting display on an off by a set of rules. The software uses the previous day's sunrise and sunset data, via a light sensor, to predict the next day's times. It is not a "real Time Clock" but it never needs (manual) adjusting.
Depending on your skill level, you could incorporate a Real Time Clock if you wanted to.