This answer is not addressed to an individual, Salome. the mother who asks the question, but to a group. So Jesus assumes she is a front for the brothers. The last phrase is addressed to them.
Children drank wine mixed in their water from a very early age in Jesus's society, but strong drink was limited to adults. Everyone would have had the experience of being told that they were not strong enough to drink from an adult's cup. Jesus saw the apostle's as his children.
For a king, the most obvious meaning was that food tasters first drank from their cup to see if the drink was poisoned. There is certainly a sense of that here, but the issue here is having the ability or the strength to do this. While it could be that food tasters were stronger than regular people so they could survive poison, there is more common experience in the time of Jesus's.
Christ's Words in Matthew as a Guide to 40 Days of Prayer.