Does size limit intelligence?
Yes, but the limits are not where many assume them to be.
Older work on the subject relying on brain size (weight), body to brain weight ratio or complexity (all those little wrinkles & folds) has been superseded by more recent work.
Intelligence appears to be limited by 'total processing power, seems it's the number of neurons & connections between neurons (so much as processing power & memory in computers, unsurprising then when you think about it) rather than it's physical size but there are physical limits to how small you can make a neuron, how closely you can pack them & how many connections you can cram in inevitably leading to limits on the physical size of a brain that has 'intelligence' of any specific level.
Some links, the elephant brain & birds forebrains, as you can see current thinking is that number of neurons & their connections in higher thought centres (not the whole brain) is where intelligence is.
So you can't just make a brain smaller & smaller & retain the same level of intelligence, eventually you reach a point where you have to begin shedding neurons (& intelligence) to get any smaller.
The smallest brains with the highest intelligence by weight (& the highest neuron count by weight) known in nature are birds (unsurprising as they need to keep weight down for flight, so any tricks there are to achieve more with less for brains will have been heavily selected for by evolutionary pressures), they have more neurons per gram of brain than in mammals.
It's not unlikely birds are pretty close to having got as close to as small as you can go for the brains' weight to intelligence ratio given the evolutionary pressures they're under to cut weight for flight & (very rough unscientific) extrapolations from a Ravens 15g brain suggest something in the region of a 45g bird brain 'might' plausible be as intelligent as a human, that's likely as small as you can go & have adult human intelligence (that of course is just the brain, you'll need a body around it too).
But it will also depend on how much of that 45g brain is given over to higher reasoning rather than other brain functions like processing sensory information & motor control of course.
"very rough unscientific" aka, 'mine'.
A 45g bird brain is speculative & there may be reasons birds techniques won't scale easily to larger ones, cooling requirements of large brains may mean you just can't pack neurons as tightly for one.
Some of the heaviest brains in birds are probably found in macaws, the hyacinth macaws is 24.7 g
Could a microorganism achieve human-like intelligence?
No, there is absolutely no way you can achieve that without magic, see above.
They'd have to organise themselves into some sort of colony or modular organism to achieve the necessary neuron count but then you won't really have a microorganism any more.