Learn Finnish: Hän on ensimmäinen asiakas.

Just like in English, the Finnish word for first (ensimmäinen) is a special word that does not look like the number one (yksi). The word for second (toinen) is also irregular. After that, ordinal numbers start to follow a more predictable pattern.

How do you write first using a digit instead of spelling it out?

In Finnish, you turn a digit into an ordinal number by adding a period after it. To write the first customer, you would write 1. asiakas.

Does ensimmäinen need to match the case of the noun it describes?

Yes, ordinal numbers act exactly like adjectives. Because asiakas is in the basic nominative case here, ensimmäinen is also in the nominative. If you wanted to say with the first customer, both words would need to change their endings.

I sometimes hear people say eka. Is that related to ensimmäinen?

Yes, eka is the very common spoken-language abbreviation of ensimmäinen. In casual conversation, it is completely natural to say Hän on eka asiakas, but you should use ensimmäinen in formal situations and writing.