Is following statement correct?
Type-up (hyphenated) and spellcheck is one word.

I have never seen "type-up" as a single word. Sounds a bit awkward.

"Spellcheck" should be written as a hyphenated word, i.e., spell-check, according to my Webster's Collegiate Eleventh Edition.

And if you are considering "type-up" and "spellcheck" as separate entities, then the plural should be used: "are one word."