The character trait that best describes Framton Nuttel is neurotic. In the second paragraph the author writes:
Privately he doubted more than ever whether these formal visits on a succession of total strangers would do much towards helping the nerve cure which he was supposed to be undergoing.
Saki intentionally contrasts him with the girl playing hostess by describing her as
. . . a very self-possessed young lady of fifteen
Her self-possession makes Framton seem all the more nervous, while his nervousness makes her seem all the more self-possessed.
Then when Mrs. Sappleton appears, Framton explains in one paragraph what his problem is. This is all the information the reader will get about his condition and virtually all that the reader will get about his character.
"The doctors agree in ordering me complete rest, an absence of mental excitement, and avoidance of anything in the nature of violent physical exercise," announced Framton, who laboured under the tolerably wide-spread delusion that total strangers and chance acquaintances are hungry for the least detail of one's ailments and infirmities, their cause and cure. "On the matter of diet they are not so much in agreement," he continued.
Framton seems to have enough money to be able to consult several doctors on a regular basis and to take time off from whatever work he does, if any, in order to spend time vegetating in the country. Saki created this character to be the perfect victim for the mischievous young Vera. Framton is going to experience just the opposite of what the doctors have advised him to do. He will probably not have "complete rest" for a long time after his encounter with the supernatural. He will get plenty of "mental excitement" when the three "ghosts" appear heading towards the open window carrying guns. He will also get a great deal of "violent physical exercise" when he flees.
Framton grabbed wildly at his stick and hat; the hall-door, the gravel-drive, and the front gate were dimly-noted stages in his headlong retreat. A cyclist coming along the road had to run into the hedge to avoid an imminent collision.
The sentence about the cyclist coming down the road creates an impression of Framton running for his life for several miles back along the country road. He would have no means of transportation except his own legs, especially with night coming on. Perhaps the frightening experience and the violent exercise would be good for him. For one thing, he might realize after running for several miles that he actually felt better than he had in years, and he might begin to suspect that the doctors were all wrong.
In that case, Vera might have been doing him a favor by concocting her ghost story. Franton might very likely make some inquiries about the Sappletons and find out that he had been the victim of an elaborate practical joke. In that case, he might be able to laugh at himself, which would be a good way to cure his neurosis.
Saki himself seems to have been a vigorous, self-reliant person. He fought in the trenches during World War I, even though he was in his early forties and not subject to conscription. He probably had little sympathy for neurotic, dependent men like Framton Nuttle or for the doctors who all had different theories about how to diagnose and prescribe for them.