On a planet the same as Earth, with no other sapient creatures, how could a Pteropus evolve sapience?
The same way we did?
Intelligence (and as a corollary, sapience) is an evolved trait. This means that at some point there must have been a pressure on us to evolve it. If these bats live in a world in which bats with smarter brains (please note that I didn't say bigger here, as they aren't necessarily correlated) are more successful, then eventually they're either going to reach a plateau where they're successful enough that the added intelligence isn't a defining factor in their breeding habits, or they're going to continue to become more intelligent.
So what does this mean for our bats?
1: There needs to be a reason that smarter bats outperform dumber bats. Imagine smart bat A, who (due to a quirk of his echolocation processing centres) has developed the ability to throw out disorienting echolocation clicks that throw off his competitors. He gets more fruit.
2: This needs to continue to be a pressure even when the bats begin to gain intelligence. Smart Bat A's descendants become the majority of bats, and start using this trick on each other. Bats who can figure out which are their clicks and which are their opponents get more fruit. This not only maintains the pressure to keep getting smarter, but also starts to develop a sense of 'self' (this is my echoclick, that is his echoclick. I am not him. I'm me.)
3: This needs to develop something in the bats brain that is capable of forming abstract patterns. For example: If I pull on vine X then fruit Y gets closer to me, but if I pull on vine Z then no fruit comes my way. This is the tricky step, but if you have a society of bats with a pressure upon them to become smarter in order to survive or breed then this will eventually happen. It's provable that many of the Corvid species exhibit this kind of behaviour, and they're pretty small for their brains.
At this point you're already well on the way to sapience, though not necessarily tool using. I'm going to take a moment to point out here that if the advantage of being smarter is outweighed by the extra weight in the brain it will be selected against, so either this is going to end up with a race of big brained, strong bats or some incredibly efficiently packed brains in smaller bats. I don't doubt the ability of a suitably long evolutionary path to deliver either, as nature is remarkably good at getting it's own way sometimes.
Once you have pattern recognising bats with some sense of self and a continuing pressure for smart bats, you will eventually end up with sapient bats using tools. From there the world is your oyster. Well, it's your oyster if the bats learn to fish...
Oh, and watch out for the smart monkeys.
I disagree with the notion that bats or flying things more generally are bad choices for sapience. I am going to suggest that bats are excellent candidates for a hive mind type sapience.
Firstly, read this about bats:
"Crucially, researchers have found that the most significant factor in determining relative brain size is often evolutionary pressure on body size, and not brain size.
For example, the evolutionary history of bats reveals they decreased body size much faster than brain size, leading to an increase in relative brain size. As a result, small bats were able to evolve improved flying maneuverability while maintaining the brainpower to handle foraging in cluttered environments.
This shows that relative brain size cannot be used unequivocally as evidence of selection for intelligence. "
Secondly, read this about bats:
“These findings upset the notion that only humans use different sides of their brains to distinguish different aspects of sound,” says the study’s senior author, Stuart Washington, PhD, a neuroscientist at Georgetown.
Washington says the findings of asymmetrical sound processing in both human and bat brains make evolutionary sense.
Now, the fact that bats are using a left/right hemisphere arrangement to process sounds/sonar when even primates don't all do that suggests based on circumstantial evidence that they are a premier choice for sapience.
However, the other responses so far are correct that flying animals of this size are going to be seriously challenged to develop sufficient neural processing power to have sapience in one bat. But take 1000 bats with all the neural architecture and 'vocal tracts' required for language already in place and give them some evolutionary nudge and I think you have an extremely plausible sapient swarm. And a sapient flock of bats is kind of even more interesting a concept than a sapient bat, singular.
SJuan76 has stated the most important thing already, so I would just explain it so as to emphasize it.
If you are talking about bats which may not have the ability to fly, then yes, you can have them. But if you want bats which can fly and are also have human-level of intelligence, it is still possible, but will take the form of some extremely outlandish and bizarre forms.
First off, I ask you this question: what is sentience (in biological terms)? What are the biological requirements of sentience? If you mean a bizarrely large body to brain ratio (precisely what we humans have) then we can start building a model around it.
Basically we are going to try and create a bat which has a very large brain for its body size. Some studies based on pterosaur models indicate that the largest weight you can have and still be able to fly, is ~75 kgs (for monsters like Hatzegopteryx and Quetzelcoatlus). This can easily accommodate a 2kg brain and a large giant head.
What is not so clear is why would a bat evolve sapience? As in, which evolutionary pressure would lead it to a higher brain size. Take our home planet for an example. The vertebrates began as fish, some 520 million years ago. These had all the prerequisites required for a very smart, sentient being: a defined brain and spinal cord. Yet we find no philosophical genus of fish with an absolutely amazing brain size (even for monsters like Dunkleosteus and Megalodon). Some 350 million years ago, some disturbed, lonely fish got bored of water-life and crawled ashore to take the form of amphibians. Even for creatures such as Prionosuchus, we find no evidence for a larger brain or extraordinary intellect. Then soonish enough, some amphibian groups got bored of having to lay eggs in water and morphed into reptiles. Within all the groups of dinosaurs and ichthyosaurs and plesiosaurs and pterosaurs and the rest of weird assortment of reptiles which ruled the mesozoic times, we find no evidence for a truly large brain case.
Mammals originated sometime near the dinosaurs (about 220 million years ago). From their origin 220 years ago to as long as yesterday (just 7 million years ago), there was no mammal with an amazing brain size. We the modern humans appeared on the face of earth just 0.2 million years ago and ... well if you compare your body to brain ratio with other creatures, you'd be literally astounded.
So, knowing that for all those 521.3 million years since the origin of vertebrates, there has been no "sapient" being in the history of life on earth. How exactly are you going to explain their origin on another earth-like planet?
Furthermore, dependence on brain grows as dependence on physical skills decreases. For a creature which is happy flying around, eating insects/fruits, there is no real need to get brainy.
What I mean to say in all this long lecture is, that creatures with large brains capable of super-intelligence are physically possible, but the process leading to such large brains in these creatures is extremely, extremely unlike to trigger.
Due to the weight and power requirement of a human brain as SJuan76 mentioned, you are going to be hard pressed to get enough lift to carry such an organism. So lets explore what you would need to lighten for this mutated bat to fly.
- A lighter and more efficient heart - after all a bigger brain will require more oxygen to stimulate the brain cells
- A quick digestive system - this will feed into the next system, but if the bat can consume, extract and excrete its food in a short time this will cut down the weight of the animal so it can still fly
- An Ability to break down food external of its body - if the bat could break down at least part of its meal outside of its body by processing the food, i.e. cooking, the bat could save digestive energy and extract more with its small digestive system
- Losing its teeth - to save weight where possible, and because the bat could prepare its food externally, the bat could lose its teeth
- Low food scarcity - this animal will need to consume food rapidly to still be light enough to fly and power such an advanced brain, so it will need to be either fulfilling one of three tasks, hunting, cooking and then eating, if no food is easily found this creature will perish
I am sure there are dozens of other ways to reduce body matter, but you can start with this list. This creature will probably be incredibly aggressive if carnivorous, vying for dominance over its food source against rival smart-bats and other predators.