Does ATP hydrolysis reaction have a high or low activation energy?
High and low are not very descriptive since they are relative. ATP hydrolysis may have a high activation energy compared to some reactions and low when compared to others.  The important point is that the activation energy is sufficiently high enough such that ATP is not rapidly hydrolysed under physiological conditions before it can do useful work. In other words ATP hydrolysis is kinetically unfavourable but thermodynamically favourable. The reaction rate is increased by enzymes (by lowering the activation energy), which couple ATP hydrolysis to other processes to do useful work.

Also, a note on reaction spontaneity: it has nothing to do with activation energy but rather depends on the change in free energy; ATP hydrolysis is spontaneous because it is exergonic.