Yes, praying is precisely what you could be doing. But even more specifically, you should be meditating.
In some of the old prayerbooks, you’ll notice that each part of the Mass has a particular, shall we say, “theme” associated with it; and this theme is then developed with prayers and devotions that can be utilized to meditate.
A good way to offer a simple distinction is to say that the Old Mass requires mental, meditative prayer from the congregation, whilst the New Mass is more focused on their being involved in vocal prayer. That is how their spiritualities differ—and how their forms of active participation are most easily understood.
Think of it like the Rosary mysteries: on the one hand, there are the Hail Marys and Our Fathers … but, on the other hand, there is also the meditation that goes along with it. That meditation is the spirit you should bring to the Old Mass. Not that you can’t bring it to the New Mass. But you won’t understand the essence of the Old Mass without it. Each part of the Old Mass is like a mystery of the Rosary; Father recites the “Hail Marys” and we meditate.
