Learn Polish: Ten dom wygląda na stary.

In Polish, to express that something gives the impression of being a certain way, we often use the verb wyglądać (to look) plus the preposition na followed by an adjective. It translates roughly to "looks to be" a certain way. If you drop the na, you have to change the grammar!

What case is the adjective stary taking here?

It is in the accusative case. The phrase wyglądać na always requires the accusative. Because dom (house) is a masculine inanimate noun, its accusative form looks exactly the same as its basic dictionary form (the nominative).

How do I say "looks old" without using na?

You can drop na, but then you must use an adverb instead of an adjective. You would say Ten dom wygląda staro. Both versions are very common and mean practically the same thing, though wygląda na stary sounds a bit more like an assessment or guess.

Why does the sentence start with Ten instead of To?

Ten is the masculine word for "this", which must agree with the masculine noun dom. You use To when pointing out a neuter noun (like to drzewo - this tree) or when making a general "This is..." statement (like To jest stary dom - This is an old house).