It depends to a certain extent where you are. I will discuss the USA and UK. This is a tricky topic partly because the advice varies depending on where you live and because the internet is not great it providing concise information on this subject.

USA
Background info: As you probably know, it is obligatory to vaccinate your cat against rabies in the USA. No vaccine is 100% effective but the rabies vaccine is extremely successful in protecting cats against the infection. Mercury News states: “Bats can have rabies, but it is extremely rare for that to impact humans. The chance of getting rabies from a bat is very small — the Centers for Disease Control and Prevention says there are one or two cases a year in the U.S.”
My research indicates that one tenth of one percent of bats in the USA have rabies but rabies is endemic in bats in the USA and as I understand it, they are the primary vector in transmitting rabies to other animals and people on this continent. But one 10th of 1% is one in 1,000. Therefore, if your cat catches a bat there is a very low chance that the disease will be transmitted to your cat and in any case, they are inoculated.
My understanding is that you should contact your veterinarian and as your cat has caught the bat you can take it to your vet to check whether it has rabies. This should obviously be done with caution wearing gloves.
You could then isolate your cat and watch and wait. Your veterinarian might recommend that a booster vaccination is given to your cat if the bat is found to be carrying rabies.
A study published in 1993, “Bats, Cats, and Rabies in an Urban Community”, states that if a bat bites a person, the rabies prophylaxis (post-infection vaccination) should be administered as soon as possible after the incident while tests are carried out on the bat to check for rabies. This gives us an indication as to the potential seriousness.
Rabies vaccination is not mandatory for people in the USA but it is for their pets.
That’s my interpretation of what might happen in the USA. I would appreciate input from visitors.

RELATED: USA – Cats – Rabies. A discussion
UK
You don’t have to have your cat vaccinated against rabies in the UK unless you are taking them abroad when it becomes obligatory. Rabies does not exist in the UK among people but bat rabies can be found in the UK. The risk of a catching rabies from a bat in the UK is extremely low but it is possible it seems to me.
The best thing to do, I think is to watch and wait. You watch your cat and if the bat is available then you can take it to a veterinarian to test for rabies as mentioned. You might want to isolate your cat pending the outcome of the test. The likely outcome is that nothing happens and the matter will be over.
An interesting aspect of this discussion is that the Bat Conservation Trust estimates that 250,000 bats are killed by cats every year. They think that this figure is a huge underestimate.
This indicates that many indoor/outdoor domestic cats are catching bats, almost certainly without the knowledge of their owner. Nothing happens in terms of the transmission of rabies to the cat because if it did there would be many reports about people getting rabies from their cat. This is not happening. This must be indicative of the very low risk in the UK of a cat getting rabies from a bat.
The problem with rabies is that it is fatal in people which creates an enormous fear of this disease which can lead to animal cruelty.
Cruel outcomes
I have read stories of stray cats in the US who have attacked people for whatever reason being trapped, killed and beheaded and an autopsy carried out on their brain. This is out of fear that an apparently unprovoked attack is caused by rabies. But the reason might simply be because the cat is fearful of a stranger or a person provokes a cat to attack and defend themselves. But sometimes questions aren’t asked and because it is “only a cat” the simple thing is to kill ‘it’ and carry out a necropsy. I don’t like it, but I understand the fear of rabies.
Below are some more articles on rabies in relation to domestic and stray cats.

Nebraska rabies scare is about raccoons not about domestic cats as inferred by the Daily Mail

Understanding the importation requirements for a domestic cat when travelling from Canada to the UK

Taking your Savannah cat (and any domestic cat) from the USA to the UK

Infographic on rabies in domestic cats (USA)
Please search using the search box at the top of the site. You are bound to find what you are looking for.
Note: sources for news articles are carefully selected but the news is often not independently verified.
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