There are a number of sources that point to the fact that Shem did indeed live a long life and also lived to see some of the personalities that you mention above.
Firstly by way of introduction, the Gemara in Bava Basra 121b highlights Shem as one of seven people, whose longevity in physical years collectively spanned the history of mankind. It writes there:
תָּנוּ רַבָּנַן שִׁבְעָה קִפְּלוּ אֶת כָּל הָעוֹלָם כּוּלּוֹ מְתוּשֶׁלַח רָאָה אָדָם שֵׁם רָאָה מְתוּשֶׁלַח יַעֲקֹב רָאָה אֶת שֵׁם עַמְרָם רָאָה אֶת יַעֲקֹב אֲחִיָּה הַשִּׁילוֹנִי רָאָה אֶת עַמְרָם אֵלִיָּהוּ רָאָה אֶת אֲחִיָּה הַשִּׁילוֹנִי וַעֲדַיִין קַיָּים
§ After discussing the generation of those who died in the wilderness, the Gemara mentions a tradition that relates to that generation. The Sages taught: Seven people spanned in their lifetimes the whole world in its entirety, i.e., their lives have spanned all of human history. Methuselah saw Adam in his lifetime; Shem saw Methuselah; Jacob saw Shem; Amram saw Jacob; Ahijah the Shilonite saw Amram; Elijah saw Ahijah the Shilonite; and Elijah is still alive. (Sefaria translation & notation)
Indeed with regards to this point, the Seder Olam Rabbah 1 notes:
ושם שמש את מתושלח צ"ח שנה
And Shem served Mesushelach for 98 years.
In Tanna D'vei Eliyahu Rabbah 24, it highlights a solid span of time that Shem was engaged in nevuah (prophecy), trying to curb the ways of the ovdei avoda zara (idol worshippers):
ולא זו בלבד אלא שם הגדול בן נח שנתנבא ארבע מאות שנה על כל אומות העולם ולא קבלו ממנו
And not only this, but the great Shem, son of Noach, prophesied for 400 years to all the nations of the world and they did not accept him.
As far as meeting Avraham, it is clear that they enjoyed several interactions and @Harel13 has already alluded to some of these in the comments above. Another example is that Avraham sent for Shem when he required bris milah (circumcision) and Shem performed the bris on both Avraham and his son Yishmael (refer to Yalkut Shimoni, Bereishis 80)
Shem definitely outlived Avraham, and the Midrash in Bereishis Rabbah 62 mentions the fact that he helped bury him. The Midrash there writes:
אָמַר רַבִּי שְׁמוּאֵל בַּר נַחְמָן שֵׁם וְעֵבֶר הָיוּ מְהַלְּכִין לִפְנֵי מִטָּתוֹ וְרָאוּ גַּבָּהּ מָקוֹם מֻפְנֶה לְאָבִינוּ אַבְרָהָם וְקָבְרוּ אוֹתוֹ בִּדְיוֹטְרִין שֶׁלּוֹ בַּמָּקוֹם הַמּוּכָן וּמְזֻמָּן לוֹ
Rabbi Shmuel son of Nachman said: Shem and Eiver were walking before his (Avraham's) bier, and they saw that the space (next to Sarah's grave) was free for Avraham, and they buried him in the spot that had been destined (lit. prepared) for him.
The Midrash HaGadol, Bereshis 11:11 definitively writes that he lived through a substantial part of Yitzchak's life too and the first third of Yaakov's life:
ויחי שם - חיה שם עד שעבר מימי יצחק מאה ועשר שנים והיה יעקב באותו הפרק בן חמשים שנה ובאותו זמן מת שם בן נח
And Shem lived - Shem lived until 110 year's of Yitzchak's life had passed, and Yaakov during this period was 50 years old, and at that time Shem the son of Noach died.
So to return to the question, yes - Shem lived in physical years for this amount of time, during which time he outlived Avraham, and lived whilst both Yitzchak and Yaakov were alive. The Seder Hadoros HaKotzair p.7 places the year of birth at 1558 and his death at 2158.