You may be interested in the travels of Faxecura, a Japanese envoy from daimiyo Date Masamune, who was sent on an embassy to open trade relations with the European powers during the Age of Discovery. Faxecura's embassy's descendants still live in Seville province, Spain, under the surname "Japón," meaning "Japanese." I don't believe that Faxecura left any record of his impressions, though, and his embassy was ultimately a failure. The Faxecura embassy coincided with the beginning of Japanese isolationism, as well as Japanese suppression of Christianity, and as such, was the last great embassy of the Japanese between 1614 and the opening of Japan by the American navy in 1854.

You could look at Boxer: The Christian Century in Japan, 1549–1650, for a good overview.

One of the first in an official capacity seems to have been Gan Ying who was sent to Rome by his commander Ban Chao in 97 CE.

Gan Ying traveled through the Tarim Basin into the Arsacid Empire but only got as far as either the Black Sea or the Persian Gulf before turning back after being warned by Parthian sailors that

"The ocean is huge. Those making the round trip can do it in three months if the winds are favourable. However, if you encounter winds that delay you, it can take two years. That is why all the men who go by sea take stores for three years. The vast ocean urges men to think of their country, and get homesick, and some of them die."

So he didn't make it to Western Europe or ever the Mediterranean and his impressions of Rome are vague and gleaned probably from local Parthians. Still rather interesting.

Batu Kahn made an official visit to Poland in the mid 1200's. I'm not sure if that is eastern enough. It sounds like he enjoyed it.

The earliest I know of was Rabban bar Sauma, a Turkic Nestorian Christian born in modern-day Beijing. He traveled to Western Europe as an envoy of the Mongolian Empire in the 13th century. His purpose was to form an alliance between the Mongol Empire and Frankish Crusaders against the Islamic Caliphates. This effort proved futile, but he did write a detailed account of his travels.

You can find an English translation of Sauma's accounts at http://www.aina.org/books/mokk/mokk.htm.