I want maximum 5V output from an Op-Amp. Is it a good idea to use a supply voltage for the Op-Amp of more than 5V?
You have two problems. Firstly, you are quite correct, the output voltage achievable from a 5 volt rail is limited to circa 4.25 volts with a 1 mA load and, to improve this you can raise the positive supply rail a volt or so.

The 2nd problem is that the input common mode range is limited to only 4 volts (page 6 of data sheet) and, given that you are operating the op-amp as a unity gain buffer, this is likely to be the problem actually seen when powering at +5 volts DC. However, this will improve if you run at a higher supply.


  And when I use a higher supply voltage for the Op-Amp I have to make
  sure that the output voltage is never higher than 5V. Because I don't
  want anything higher and a higher output might destroy the ADC which
  measures the output voltage.


You might be able to use a series resistor in your ADC input to limit the current to an acceptable value. Read the ADC data sheet to check. Else, I would use a schottky diode from the op-amp output to the local 5 volt rail of the ADC. A zener could be used but they tend to be a little leaky below their clamping point and may not quite be good enough for what you want.

Also, the op-amp you have chosen is not that great on input offset voltage but, if you are content with a standing error of up to typically +/- 1 mV then that's OK.