I know one person who read (at least parts of) Carr's Synopsis when they were young; they are now an exceptionally strong mathematician, but I don't think there is a causal relationship. Rather, there is a correlation between their early interest in mathematics and their independence, and their interest in reading unusual original sources like Carr's Synopsis.
Regarding the question of reading original sources (especially old ones): I think this tends to be discouraged, and the reasons are related to Qiaochu's comment --- it is hard to both learn everything that is necessary to do modern mathematics and spend time reading all the original sources. On the other
hand there are advantages that can come from reading older sources that are often not mentioned: for example, one can find points of view or topics or techniques that are no longer emphasized, or are even forgotten, which can sometimes be helpful in a very specific way, and other times helpful in a more general inspirational sense.
To speak of myself for a moment, when I was learning mathematics I read many older and somewhat unusual sources, and I think that there are advantages to this, as well as the disadvantages mentioned above. I think that I have an above average interest in the history of mathematics, and of the historical development of mathematical ideas, and so it was enjoyable and profitable for me to do this. But if I had a different mathematical temperament this wouldn't have been a sensible choice; it was a reflection of my own mathematical personality.
If you find it interesting to read older sources, or you think it might be interesting to try, then I would suggest that you just look at them and see what you get out of it. Just bear in mind that you won't be able to learn modern mathematics in this way, so you wouldn't want such reading to be your only reading by any means; rather, you might want to regard it as a supplement to your regular study of modern expositions.
As for learning specifically about Ramanujan's work, the best place to start is Hardy's book on Ramanujan. This is a wonderful book.
On the other hand, to approach Ramanujan's legacy in a modern context, the best thing to do is to learn the theory of modular forms. For this, Serre's Course in arithmetic is a very good source. (Serre's book has three parts which are only loosely connected. So you can read the section on modular forms without having read the other sections carefully beforehand.)