Rabbi Ahron Marcus discusses this issue shortly in his book "Barzilai" (pg. 313 in the Hebrew edition):
"קושי שלישי שעורר אופרט...בדניאל א, 21 נאמר: "ויהי דניאל עד שנת אחת לכורש המלך." ואילו בפרק י מנבא דניאל את נבואתו האחרונה "בשנת שלוש לכורש מלך פרס". אופרט לא שם לב, כי בדניאל א', 21 נאמר "ויהי", לא "ויחי". אילו היה כתוב "ויחי", היה הדבר נוגד את רוח השפה העברית בהקשר זה...כל תלמיד בחדר מבין את הכתוב, כי דניאל התמיד במשרתו בבבל עד לעלייתו של כורש לשלטון, ועבר אחרי כן לפרס, שם ראה את חזונו האחרון "על יד הנהר הגדול הוא חדקל" (דניאל י, 4)..."
Translation: "A third difficulty raised by Oppert...in Daniel 1:21 it says: "Daniel was there until the first year of King Cyrus." but in chapter 10 Daniel received his final prophecy "In the third year of Cyrus king of Persia". Oppert didn't notice that in Daniel 1:21 it says "[he] was" [ויהי], not [he] lived" [ויחי]. Were it so that "[he] lived" was written, this would have contradicted the spirit of the Hebrew language in this context...every schoolboy understands that what's written is that Daniel continued on his duty in Babylon until the rise to power of Cyrus, and then he moved to Persia, and there he saw his final vision "on the bank of the great river—the Tigris" (Daniel 10:4)..."
According to Rabbi Marcus, the first verse refers to how long Daniel lived in Babylon. After that, Daniel moved to Persia (which makes sense, in light of the meimra, which appears in several places, that Daniel was Hatach of Megillat Ester) and it was there that he received his final vision.
Note: I think your question is good, which is why I feel I should try to explain why Rabbi Marcus basically laughed away the question (in fact, I didn't even write everything that he had to say to Julius Oppert). Although I haven't merited to personally know Rabbi Marcus (he died almost a century ago), I've read a few of his books and one essay of his. He seems to have seen it his duty to utterly decimate Bible Criticism, in order to sanctify the Torah and sanctify the name of Hashem. As such, whenever he discussed questions raised by Bible Critics, he would do his best to "beat them up" as much as he could. I believe that were he to be asked the same questions by people who are just trying to understand Torah, he would have replied in a wholly different manner.
The RI"D, in his commentary on Daniel, has a different explanation:
"ויהי דניאל - בחצר מלכי בבל עד - שנה אחת לכורש - והוא כורש הגדול בן אחשוורוש שבנה הבית." (Daniel 1:21)
Translation: "Daniel was there - in the court of the kings of Babylon until - the first year of Cyrus - and he is Cyrus the Greater son of Achashverosh who built the Temple.
"בשנת שלוש לכורש מלך פרס דבר נגלה לדניאל - זהו כורש הראשון, שבשנה הראשונה למלכותו היתה הפקידה, ובשנת שלש למלכותו נגלה לדניאל זה המראה." (Daniel 10:1)
Translation: "In the third year of Cyrus king of Persia an oracle was revealed to Daniel - this is Cyrus I, that during the first year of his rule came the charge1, and in the third year of his rule this oracle was revealed to Daniel."
In other words, the RI"D thought that the Cyrus in ch. 1 refers to Daryavesh II, son of Achashverosh (and Esther), who assisted with finalizing the building of the Temple (see Ezra 5:1-6:15), while the Cyrus in ch. 10 refers to Cyrus I, who lived several years prior and was the king who first allowed the Jews to return to Judea and build the Temple (see Rosh Hashanah 3b on the names and titles of Persian kings).
1 I assume this is a reference to this: "כֹּה אָמַר כֹּרֶשׁ מֶלֶךְ פָּרַס כֹּל מַמְלְכוֹת הָאָרֶץ נָתַן לִי ה' אֱלֹקי הַשָּׁמָיִם וְהוּא פָקַד עָלַי לִבְנוֹת לוֹ בַיִת בִּירוּשָׁלִַם אֲשֶׁר בִּיהוּדָה." - "“Thus said King Cyrus of Persia: The LORD God of Heaven has given me all the kingdoms of the earth and has charged me with building Him a house in Jerusalem, which is in Judah." (Ezra 1:2)
