Is the future perfect tense still common in English? If not how could I say this sentence instead?
"I will have typed all the letters by next two hours."
As The Photon says, the future perfect is acceptable. It’s not exactly “common”, because the need for it doesn’t arise often, but there’s nothing odd or archaic about it.
The Photon also observes that you would not use the phrase by the next two hours with the future perfect. This is because the next two hours designates a two-hour timespan, not a point in time. As a time locative, by does not take a timespan but a point in time as its object. You can say by one o’clock or by Thursday or by two hours from now, because these all designate points†, but you cannot say by the next two hours or by my lifetime.
Moreover, there are peculiar rules about time locatives with perfect constructions. Perfect constructions are relative tenses: that is, a perfect names an eventuality which occurred before some particular point in time, called the ‘Reference Time’ (RT). With a present perfect, the RT is in the present, with a past perfect the RT is in the past, and with a future perfect the RT is in the future. In your example, your time locative designates that RT, a particular point in the future, two hours from now, at which you are confident you will be able to say “I have typed all the letters”.
† Granted, they are rather fuzzy points; but usually, in this sort of context, expressions like Thursday or August or the 21st century are understood to mean something like some time on/in the time or date indicated.