Palm Sunday, the Sunday prior to Resurrection Sunday, is the day Jesus enters Jerusalem on the back of a colt. Upon His entry into the city the Jews, believing He was coming to liberate them from the Romans, lined the streets in celebration waving palm branches and laying them, and in some cases their cloaks, on the ground for Jesus to ride over. They were anticipating something momentous, yet what they were anticipating and what ended up happening wasn't anywhere even close. The Jews believed He would be their savior from decades, if not centuries, of living under the oppressive rule of one after another conquering nation. Rome was just the latest of many.
But that is not what Jesus came to do. What God had planned was astronomically more important than delivering one small nation from being a scattered, oppressed people. What ended up happening on a human level was it became clear that Jesus had no intention of establishing, or reestablishing, a kingdom just for the Jews with Himself as their king, therefore, the Jewish leaders sought to discredit Jesus and had Him executed. So, on a human level, as far as the Jews were concerned, Palm Sunday was just a glaring reminder of a huge disappointment. So, there is no reason why, based on this example, the Jews should celebrate this holiday.
However, what really was going on was much more powerful and exceedingly more important. Jesus rode into Jerusalem on that day to face the culmination of, and the fulfillment of, His ministry on this earth. It was to save mankind from death and sin, a much more powerful and debilitating oppressor than the Romans could have ever hoped to have been, though they certainly did try. Palm Sunday is, in Christianity, the beginning of Passion Week, which brings the end of the season of Lent, which ends on Maundy Thursday (Holy Thursday), and includes the betrayal, death, and resurrection of Jesus Christ. This is, arguably, the one event in the history of the world, much less the Christian Church, that should be celebrated as it is the impetus for the Christian religion, and the foundation on which Christianity is built.
So, the answer to the question: Should Christians celebrate Palm Sunday? Is most certainly a big: Yes!