What is the possibility that the drink offering was poured into the altar of sacrifice?
No, it was poured into a special cup on the alter with a hole in the bottom leading to a pit underneath.
נִיסּוּךְ הַמַּיִם כֵּיצַד
With regard to the rite of water libation performed in the Temple during the Festival, how was it performed?
שְׁנֵי סְפָלִים שֶׁל כֶּסֶף הָיוּ שָׁם רַבִּי יְהוּדָה אוֹמֵר שֶׁל סִיד הָיוּ אֶלָּא שֶׁהָיוּ מוּשְׁחָרִין פְּנֵיהֶם מִפְּנֵי הַיַּיִן וּמְנוּקָּבִין כְּמִין שְׁנֵי חוֹטָמִין דַּקִּין (וְאֶחָד) מְעוּבֶּה וְאֶחָד דַּק כְּדֵי שֶׁיְּהוּ שְׁנֵיהֶם כָּלִין בְּבַת אַחַת מַעֲרָבוֹ שֶׁל מַיִם מִזְרָחוֹ שֶׁל יַיִן
There were two silver basins there into which he poured the water. Rabbi Yehuda said: They were limestone basins, but they would blacken due to the wine and therefore looked like silver. The two basins were perforated at the bottom with two thin perforated nose-like protrusions. One of the basins, used for the wine libation, had a perforation that was broad, and one, used for the water libation, had a perforation that was thin, so that the flow of both the water and the wine, which do not have the same viscosity, would conclude simultaneously. The basin to the west of the altar was for water, and the basin to the east of the altar was for wine.
תַּנְיָא אָמַר רַבִּי אֶלְעָזָר בַּר צָדוֹק לוּל קָטָן הָיָה בֵּין כֶּבֶשׁ לַמִּזְבֵּחַ בְּמַעֲרָבוֹ שֶׁל כֶּבֶשׁ וְאַחַת לְשִׁבְעִים שָׁנָה פִּרְחֵי כְהוּנָּה יוֹרְדִין לְשָׁם וּמְלַקְּטִין מִשָּׁם יַיִן קָרוּשׁ שֶׁדּוֹמֶה לְעִיגּוּלֵי דְבֵילָה
It was taught in a baraita that Rabbi Elazar bar Tzadok said: There was a small gap between the ramp and the altar west of the ramp, and once in seventy years young priests would descend there and gather from there the congealed wine left over from the libations that set over time, which resembled round cakes of dried and pressed figs.