In the sentence, the goal is to convey a feeling or state of being surprised about receiving a letter. The correct phrasing should clearly reflect this intended meaning. Analyzing the options:
doubtless be somewhat: This choice is incorrect because "doubtless" typically implies certainty, and the adverb "somewhat" does not clearly accompany the intended meaning.
doubtlessly be somewhat: This option is correct. "Doubtlessly" modifies the verb "be," indicating the speaker's expected reaction ("somewhat surprised"). The adverb "somewhat" is suitable as it qualifies the degree of surprise.
somewhat doubtlessly be: This arrangement is incorrect. The adverb "somewhat" should come after "be" to modify the level of surprise appropriately.
doubtlessly somewhat be: This structure is grammatical but awkward. The positioning of "somewhat" before "be" detracts from clarity.
Thus, the most natural and grammatically correct completion is: "You would doubtlessly be somewhat surprised to get this letter."