"The Variable Man", a novella by Philip K. Dick; available at Project Gutenberg and Librivox; first published in Space Science Fiction, September 1953, available from the Internet Archive. Here is part of the Wikipedia plot summary:
[. . . .] This is where Thomas Cole, known as The Variable Man, comes in. Cole is a man from the past, from 1913, the time just before the First World War. He is brought into the present (or future depending on perspective) as an accident via a Time Bubble that was used for research about the past. He escapes from the authority in the future and spends a lot of time running from them afterwards. It is, however, discovered that this man has a certain genius to fix things and make things work. This is because he comes out of a period of time when humans had a natural genius and an ability to invent things and to solve problems. It is at this point that the man working on the FTL (Faster Than Light) bomb realizes that The Variable Man is the only person who can make Icarus work. As a result, the engineer working on Icarus convinces The Variable Man to help them out. Icarus does eventually work, although not in the way that anyone may have wanted. Instead of emerging from FTL speed in the middle of Centarus (the sun around which the Centaurian Empire is built) and blowing it, and the surrounding Centaurian system, out of existence, it turns out that Cole transformed (or fixed) Icarus into a working hyperdrive. However the order for Terra to launch a full-scale attack against the Centaurian Empire (under the assumption that the majority of the enemy ships and planets would have been destroyed in the Icarus explosion) had already been given. The forces of Terra suffered a terrible defeat, losing many of their ships, yet due to the Variable Man having successfully wired Icarus it was now possible for Terra to travel beyond the Centaurian Empire's perimeter. Terra was no longer blocked into their tiny system, and there was no further need for war.
It could have been one of those 2 Stories in one Book (opposite sides). I probably read it in the early to mid 60s.
You probably read it in Philip K. Dick's 1957 collection The Variable Man and Other Stories.
A tinker/knife sharpener in the 30s(?) or late 1800s, had a horse drawn cart and went through a time portal (invisible?) to what could be late 1900s or more.
Thomas Cole was sharpening a knife with his whetstone when the tornado hit.
[. . . .]
All at once it was there, completely around him. Nothing but grayness. He and the cart and horses seemed to be in a calm spot in the center of the tornado. They were moving in a great silence, gray mist everywhere.
He was snatched from the year 1913:
Fredman shifted uncomfortably. "There's not much to tell. I gave the order to have the automatic setting canceled and the bubble brought back immediately. At the moment the signal reached it, the bubble was passing through the spring of 1913. As it broke loose, it tore off a piece of ground on which this person and his cart were located. The person naturally was brought up to the present, inside the bubble."
The story is set in the year 2136:
At this moment, 9:30 AM, May 7, 2136, the statistical ratio on the SRB machines stood at 21–17 on the Centauran side of the ledger. All facts considered, the odds favored a successful repulsion by Proxima Centaurus of a Terran military attack.
He was confused about a lot of things, but was able to understand how to fix anything
"I'm looking for work," Cole murmured. "Any kind of work. I can do anything, fix any kind of thing. I repair broken objects. Things that need mending." His voice trailed off uncertainly. "Anything at all."
After he fixes a broken toy "vidsender":
Reinhart and Dixon looked at each other. "This is bad," Reinhart said harshly. "He has some ability, some kind of mechanical ability. Genius, perhaps, to do a think like this. Look at the period he came from, Dixon. The early part of the twentieth century. Before the wars began. That was a unique period. There was a certain vitality, a certain ability. It was a period of incredible growth and discovery. Edison. Pasteur. Burbank. The Wright brothers. Inventions and machines. People had an uncanny ability with machines. A kind of intuition about machines—which we don't have."
[. . . .]
"This man is different. He can fix anything, do anything. He doesn't work with knowledge, with science—the classified accumulation of facts. He knows nothing. It's not in his head, a form of learning. He works by intuition—his power is in his hands, not his head. His hands! Like a painter, an artist. In his hands—and he cuts across our lives like a knife-blade."