You're asking several different questions all at once.

More specifically, the remaining Samaritans claim that they practice the actual religion of ancient Israel, and Judaism is a corruption that occurred during the Babylonian Exile. How true is this? What sort of changes did the exile bring about?

This is a highly contentious area within the academic study of Ancient Israelite religion. Whether or not the Torah (the first five books; also called the Pentateuch) were redacted before, after, or during the Babylonian Exile, for example, is quite debated. Much of the first Five Books are essentially a rule book for how to conduct Israelite religious practices, so the debate about when that was set in place.

However, it should be noted that the Samaritan Pentateuch is very close to the contemporary Masoretic text used by Jews worldwide (as well as Christian translators). There are many minor points of difference, however, many of these textual variants line up with other ancient texts we have, like the Septuagint (a 2nd century BCE translation of the Torah into Greek; often abbreviated LXX) indicating that they're not so different, and that Judaism and Samaritanism split ways after the Pentateuch was redacted.

There are, however, some major differences between Samaritan Pentateuch and the Jewish one. Most notably, whereas the Jewish Pentateuch has G-d indicating that people will build a "House of the Lord" (Temple) in "a place that [G-d] will show them", with the implication in the ears of all readers that the place that He will show us is Mt. Zion in Jerusalem (foreshadowed, for example, by the Melchizedek anointing Abram there) where Solomon eventually built his great Temple, in the Samaritan version the place is specified as Mt. Gerizim, where Samaritans still conduct sacrifices today. Whether or not the Temple was built similar to a way described in the Book of Kings (a minority Biblical ultra-minimalist scholars deny the existence of a historical Solomon; some are very prominent, however, like Israel Finkelstein), there is evidence that Jerusalem was a cultic center before the Babylonian exile. Some of the textual evidence, too, assumes are from the period of the Babylonian Exile (though we have no date from other than context) indicate that the cultic center was already established at the Temple on Mt. Zion. Psalm 137, for example, starts off:

By the rivers of Babylon there we sat down and there we wept when we remembered Zion.

This does not mean that there wasn't also a cultic center at Mt. Gerizim before the Babylonian exile--remember, that, even accepting the broadest outline of the Biblical narrative in the Book of Samuel and the Book of Kings, for most of the period from the establishment of the Kingdom of Israel to the Babylonian Exile, there were two kingdoms: Judah, in the south with its capital in Jerusalem, and Israel, in the north with its capital in Samaria. The text itself suggests that there were alternate sites of sacrifice in the Northern Kingdom. Nor does it mean that Mt. Zion was the original place of sacrifice for the Israelites. In fact, the text makes it clear that it was not. Not only was Jerusalem/Zion only conquered by David (after not only all of the Pentateuch, but also all of the Book of Joshua, the Book of Judge, and most of the Book of Samuel) and there clearly existed cultic centers of worship before that (Beth-El, for example, or Shiloh, where the Prophet Samuel was based), but even after Solomon built the Temple in Jerusalem, other cultic centers persisted. These competing centers--often referred to as "High Places"-- quite possibly were not eliminated, meaning that all sacrifices were not effectively consolidated into Jerusalem, until likely the reign of Hezekiah, and quite possibly not until the reign of his grandson Josiah (who was king very shortly before the Babylonian exile).

Would someone practicing Judaism today, if transported back to 1000 BC, even recognize the religion of the time (and likewise in the reverse)?

The only real evidence we have about religious practice in 1000 BCE is from the Torah. If the Biblical account is even vaguely accurate, an Orthodox Jew would certainly recognize things that they'd read about, but remember, this is before the building of the Temple, and at some of the High Places there might be unfamiliar aspects like Asherah poles. We don't really know. Language would obviously be different from Biblical Hebrew, but might be similar enough for a person familiar with Biblical Hebrew to get by (the languages change considerably 500 years, especially I'd guess before the advent of printing).

What about a practicing Jew merely from around 1 AD?

This would be more familiar. This is in the earliest period of rabbinic Judaism, there are almost certainly some synagogues by this point, though we don't know exactly what people did in them. Everyone would be speaking Aramaic, but many Orthodox Jews can read Aramaic as that's the primary language of the Talmud. The biggest difference between then and now, however, is that then the Temple still stood. This means that the focus of Judaism was on the Temple. There were still hereditary priests (not rabbis, Levite and Kohen priests). There were still daily sacrifices. All of that. Now, a contemporary Orthodox Jew would recognize all of that because what sacrifices were conducted when are still ready out during the times of the prayers services, we just feel we can't actual perform these sacrifices without the Temple. People have even prepared all the vessels and vestments for the restoration of sacrifices, should the Third Temple descend from Heaven and the Messianic Era begin tomorrow. As a side note, Samaritans never stopped sacrifices, but only conduct one yearly sacrifice, on Passover.

One last thing you may be interested in, the Talmud tells a story:

Rav Yehudah said in the name of Rav: When Moses ascended to the Heavenly Heights, he found the Holy One, Blessed is He, as He was sitting and attaching crowns to [some of] the letters. [That is, although the Torah was completely written, He was still adding tagin to certain letters] [Moses] said before Him, "Mast of the Universe--who is holding You back [from giving the Torah as it is]?" [God] said to him, "There is one man who is designed to exist at the end of many generations--Akiva ben Yosef [Rabbi Akiva] [and it is he] who will expound upon each and every point--heaps and heaps of halachot [laws]. [Moses] said before [God] Master of the Universe, show him to me!" God said to him, "Turn around."

[He found himself in R. Akiva's class.] Moses went and sat at the end of eight rows of students but he did not understand what they were saying. [Moses'] strength ebbed. Once they reached a certain matter, [R. Akiva's students] asked him, "Teacher from where do you know this?' [R. Akiva] replied to them, "It is a halachah [law] [transmitted orally] to Moses at Sinai." Upon hearing this, [Moses'] mind was relieved.

So even at this relatively early juncture (R. Akiva lived 40-137 CE, so we was in the generation that witnessed the Temple being destroyed; Rav Yehuda who narrates this story lived 220-299 Ce), there was recognition that there was tremendous change in Judaism--Moses, for example, did not understand what was being discussed in Akiva's classroom. However, despite the change, there was also continuity, and a strong belief that despite the changes, there was an unbroken tradition dating all the way back to Moses.