Aristotle's notion of experience is quite complex and has puzzled commentators ever since, particularly because it is not easy to reconcile differing passages in Aristotle's writings.
Met. I 981a:
[…] αποβαίνει δ` ἐπιστήμη καὶ τέχνη διὰ τῆς ἐμπειρίας τοῖς ἀνθρώποις.
[…] but really science and art come to men through experience.
The central point to remember in this particular case is that for Aristotle
empirical knowledge does not stand in opposition to
rational knowledge, but that the former is a part of the latter. For Aristotle, to experience something presupposes a rational capacity.
Aristotle is not always consistent on this point, and thus some passages suggest that experience might be a non-rational capacity. There is a helpful paper on this, which might help you in your reading of Met I:
Aristotle's notion of experience plays an important role in his epistemology as the link between perception and memory on the one side, and higher cognitive capacities on the other side. However, Aristotle does not say much about it, and what he does say seems inconsistent. Notably, some passages suggest that it is a non-rational capacity, others that it is a rational capacity and that it provides the principles of science. This paper presents a unitary account of experience. It explains how experience grows from perception and memory into a rational capacity, and in what way it provides the principles.
(Pavel Gregorić, Filip Grgić, "Aristotle’s Notion of Experience", AfGPh 88 (2006), 3)
In the passage you quoted, Aristotle hints at the process of epagoge, customarily translated as "induction", through which humans can get from sense-perception to grasping universals. Cf. this more detailed account by Aristotle:
Therefore we must possess a capacity of some sort, but not such as to rank higher in accuracy than these developed states. And this at least is an obvious characteristic of all animals, for they possess a congenital discriminative capacity which is called sense-perception. But though sense-perception is innate in all animals, in some the sense-impression comes to persist, in others it does not. So animals in which this persistence does not come to be have either no knowledge at all outside the act of perceiving, or no knowledge of objects of which no impression persists; animals in which it does come into being have perception and can continue to retain the sense-impression in the soul: and when such persistence is frequently repeated a further distinction at once arises between those which out of the persistence of such sense-impressions develop a power of systematizing them and those which do not. So out of sense-perception comes to be what we call memory, and out of frequently repeated memories of the same thing develops experience; for a number of memories constitute a single experience. From experience again-i.e. from the universal now stabilized in its entirety within the soul, the one beside the many which is a single identity within them all-originate the skill of the craftsman and the knowledge of the man of science, skill in the sphere of coming to be and science in the sphere of being.
(Posterior Analytics, II.19)
This blog post Aristotle's View of Induction: A Summary gives an accessible overview on the subject matter.
So, to answer your questions:
1. Are non-empirical sciences part of epistéme?
They certainly are. In Aristotle's conceptual division of epistéme both empirical and non-empirical sciences are part of the branch of theoretical sciences (the other branches being practical and productive sciences).
One part is the process of epagoge which you quoted above. However, this is not enough to grasp the necessary causes in nature, which, according to Aristotle, knowledge of the world is really about. Aristotle's definition of knowledge is very strict: We only attain knowledge in the full sense when we know
the cause why the thing is, that it is the cause of this, and that this cannot be otherwise. (Posterior Analytics I.2)
This second part can only achieved through deductive reasoning, which Aristotelian logic, his account of syllogism is concerned with. This reasoning is necessary to attain demonstration in the strict sense, and it features in all of Aristotle's theoretical sciences, both in physics and in mathematics.
2. Did Aristotle know about non-empirical science?
Aristotle did know about Euclid and the current developments in mathematics. The available evidence is referenced in a related question Which work by Aristotle features the most references to Euclidean geometry?.