First off, 1000 CE would be the very early Song dynasty (the Northern Song 北宋) in China, with the Liao (遼朝) / Khitan state still holding sway in the northern regions. The capital of the Song in this period was Bianjing (汴京), now the city of Kaifeng (開封) in Henan Province. In the words of Pulleyblank (1991):
Unfortunately, evidence for the language of Northern Song and its northern and southern successors, Jin and Southern Song, is sporadic and fragmentary.
However, there is enough long-lasting data (i.e. books and records) for our 21st century language learning purposes, such that it could be reconstructed. But it depends on what you are reconstructing, and what you think your reconstruction represents.
Note: the language of the Song Dynasty is conventionally known as Late Middle Chinese (晚期中古漢語), dated rather vaguely to the 10-12th centuries CE (hence including the late Tang). Caveats about topolect and time periods / generations apply.
The lack of segmental information inherent in the non-alphabetical system of Chinese characters makes it challenge to reconstruct sound. However, phonological research was at an (all-time?) peak through this period, as a consequence of the imperial examination and the cultural love of poetry, and the exposure to Buddhism and hence Indian lingustics.
Hence, some very precious sources have been passed down to us, enough for a good set of reconstructions of Middle Chinese to flourish. The primary one is the Qieyun (切韻), dated to the early 7th century, in the Sui dynasty (隋朝), which gives us a very good base for Early Middle Chinese. More relevant to the Song dynasty is the Guangyun (廣韻), which is now available digitally.
Added to these, what I would call "the key to unlocking the dictionaries", are the rime tables, most notably for the Song dynasty the Yunjing (韻鏡), which is one of the oldest ones. By classifying the tones (which are the easiest to understand!), rimes (= medial glide + vowel nucleus + final consonant) into a table, with the names of the initial consonants classified too, it makes understanding the older dictionaries much easier.
The fanqie system employed across historical Chinese linguistics is perhaps not the most intuitive for a modern language learner, and there are long-term debates in some of the medials (e.g. "What are the chongniu (重紐)?"), but it is enough to know how virtually any of the characters could be reasonably pronounced. Major systems include Baxter's and Li's.
Lexicon & grammar
Knowing lexis and grammar simply depends on having a good body of data. However, "good" means different things to different people.
This something that Classicists also struggle with. Do we really think that anyone spoke like Cicero? Or that the language on the streets of Athens was ever the same as Attic? How realistic was Caesar's speech in the Campaigns as a reflection of his own daily tongue? Was Paul of Tarsus's use of Koine Greek equivalent to its use in the communities he visited? These questions all have parallels across any historically documented language, even as recent as the Victorian Age for English.
With the strongly diglossic situation across all of the East Asian sprachbund, caused by the much more culturally and administratively centralised nature of education and literacy of the Confucian canon, the difference between what one of the Song dynasty literati wrote and what he spoke would have been rather large. The dominance of Literary Chinese was probably not challenged throughout the period, as VH Mair's entry for the Encyclopedia of Buddhism (2004) says:
The mainstream Confucian literati never accepted anything other than Literary Sinitic as a legitimate medium for writing. To them the vernacular was crude and vulgar, beneath the dignity of a gentleman to contemplate.
Nonetheless, literacy was not limited to the literati, and so vernacular forms of Middle Chinese were not unknown in this era. As Qing dynasty chronicler Jiang Fan (江藩) stated in the 《國朝宋學淵源記》, on the subject of Chan (Zen) Buddhists and Song dynasty Confucians:
The Chan School had Yulu and Song Confucians also had Yulu; the Chan School used vernacular language in their Yulu and Song Confucians also used vernacular language in their Yulu
The Yulu are collections of sayings, and hence for the Buddhists were strongly inclined towards the vernacular (lit. street language). Perhaps surprisingly, the Neo-Confucians also had substantial vernacular influence too.
However, since most of the literature is in essence a form of Literary Chinese, one would be learning a Middle Chinese reading of Literary Chinese. Again, whether this is "authentic" Late Middle Chinese depends on your definition.
What is very different to the Mediterranean Classicists is that good primers are more commonly available in Chinese than in other European languages (although I've heard good things about Japanese and Korean primers of kanbun), and that oral reading is in the modern language of Mandarin, Cantonese (or Japanese or Korean or Vietnamese). But they do exist.
There's even "living language" versions now, where Literary Chinese is treated much like a modern foreign language guidebook rather than the grammatical primer style, starting with greetings and culture rather than the first declension and conjugation / the subject pronouns and the most common final particles. I've digitally misplaced my copy however - will update when I find it again.