First, proper names are specific enough, so they don't require an article. Which is why, for example, this phrase the Cambridge University sounds odd if left unmodified. Modify it, though, and it works;e.g., Are we talking about the Cambridge University I attended?
Second, from what I can tell based on the distribution of this name Baikonur Cosmodrome, it appears that there are two ways to express it:  with an article when the facility itself is emphasized, and  without an article when the location, the place is emphasized:
 The Baikonur Cosmodrome is the oldest and largest operational space launch facility in the world.
 In 1957, some fifty years ago, the world's first artificial satellite, Suptnik-1, was launched from Baikonur Cosmodrome. Since the launch, the Cosmodrome ...
Now, that's not to say every speaker, English, Russian-English, and so on, adhere to the same distributional patterns above. But what we do know is that there is a pattern and it's more regular than irregular.
Does that help?