There is an easy answer and a hard answer. Let's start with the hard answer:
Sacred Literature of the Samaritans
From Hjelm, Ingrid. The Samaritans and Early Judaism A Literary Analysis, p 94
often-stated 'critique' of the Samaritans' rejection of the Hebrew
Bible and of their recognition of only the Pentateuch needs a
clarifying remark. The Samaritan rejection of the Hebrew Bible does
not imply that they did not develop their own traditions of chronicles
and halakhah. As Jews did not consider the Pentateuch to give answers
to all matters of life, so the Samaritans gave a certain credit to
tradition and to the interpretation of the Pentateuch.
The issue is, do we have any manuscripts of Samaritan chronicles or halakha that might date to the time of Christ? First all clearly extant Samaritan documents we have date at the earliest to the third century CE (see Hjelm's summary of extant sources from p95-103).
But at the same time, the reason why we do not see a uniquely Samaritan halakha before the third century CE is that they used a common pool of texts with other sectarian groups, choosing to ascribe cannonicity in their own way to these texts -- just as the process of cannonization was not completed for any sectarian group at the time of Christ. (The MT tradition may not have even existed until after 100 CE). Thus the Samaritans would have access to a common pool of texts and used these texts, classified as "pre-Samaritan", in the sense that the evolution of a unique textual stream that diverged from the rest of judaism was not yet complete. Here is Hjelm again (p.81)
Of importance for the relationship between pre-Samaritan and
Samaritan texts is the absence of any so-called sectarian readings in
pre-Samaritan texts. The designation 'pre-Samaritan' is therefore
based on script, expansionism, harmonization and linguistic
features. Expansionism and harmonizing tendencies, which in
pre-Samaritan texts do not bear the same characters, however, are
found in several other DSS texts and should perhaps more precisely be
labelled expansionist texts. I think it proper to argue against Tov21
that the Samaritans cannot be said to have chosen such a text, but
rather continued to use the text-type they were accustomed to. This
statement of course implies a different view of the Samaritan origins
as well as of whether texts have been expanded or shortened. It should
not go unnoticed that we do not find any Masoretic texts at Qumran
and that some of the texts called proto-Masoretic bear close
similarities to so-called expansionist texts in a manner hardly to be
distinguished from the MT Isaiah, Jeremiah and Ezekiel.22 This in fact
was confirmed also by E.Y. Kutscher, who, in his study of IQIsa23
detected a great degree of similarity between the IQIsa, LXXand SP.
All three text-types intended to remove linguistic and theological
ambiguities and seemed more suited to popular use.
In this view, the Samaritans would have access to Isaiah and other texts found in Qumram, but in the future, would not include them in their cannon once the cannonization process was complete and the idea of a Samaritan stream took over from the "pre-samaritan" stream which was non-sectarian and used by all sects.
Messiah as a popular belief
With respect to beliefs, we have to understand the belief in a Messiah was a popular belief. Many people claimed to be Messiah around that time and attracted followers from all groups.
For the Samaritans however, the basis for rejection from canonnization of existing prophets was that there was no prophet equal to Moses - except for the one prophet predicted by Moses. Here is Paolo Sacchi in History of the Second Temple Period p 30, discussing Samaritan interpretation of Deut 18.15-20:
Regarding the passage from Deuteronomy, in the intentions of the
author the word ‘prophet’ does not refer to a future figure, but the
permanent institution of Israel. This becomes clear from the context:
first it speaks of the monarchy, secondly of the priesthood, and
thirdly of prophetism. The messianic interpretation of the passage
originated with the Samaritans who, having excluded the Prophets from
the canon, interpreted the passage as though there were only two
prophets: Moses and a later one equal to Moses who would come as a
restorer.4 The great antiquity of the Samaritan conception of the
messianic figure as ‘prophet’ is extremely probable.5
Thus the Samaritans would have been looking for Messiah all along. Note that the Samaritan woman at the well says "I see you are a prophet". For a Samaritan to call someone a prophet was a different matter than for a jew to call someone a prophet.
So the combination of Samaritan's expectations of a Messianic prophet, the general popular sense of a coming Messiah, then this should be enough to justify the Samaritan's woman's faith.
If even the the Magi could travel to Bethlehem to seek Christ, it should not be hard to believe a woman in living Palestine could know about Christ, once we accept the likelihood that
Samaritans had shared access to a common pool of non-sectarian sacred texts that included Isaiah and other apocalyptic works, even if they disagreed on issues of cannonicity.
The Samaritans were looking for a true prophet to arise after Moses who would be a restorer.
There were many popular messianic movements in Palestine at the time