I would like to achieve an operation in MongoDB that would be analogous to doc.value = max(doc.value - amount, 0). I could do it by fetching document, updating its value then saving it, but is it possible with an atomic operation to avoid problems with synchronisation of parallel decrements?

It is, in fact, possible to achieve with a single operation. All you need is an aggregation pipeline inside an update operator.

Let's say you have a doc that looks like this:

{
    "key": 1,
    value: 30
}

You want to subtract x from value and if the resulting value is less than zero, set value to 0, otherwise set it to whatever value - x is. Here is an update aggregator you need. In this example I am subtracting 20 from value.

db.collection.update({
  key: 1
},
[
  {
    $set: {
      "value": {
        $cond: [
          {
            $gt: [
              {
                $subtract: [
                  "$value",
                  20
                ]
              },
              0
            ]
          },
          {
            $subtract: [
              "$value",
              20
            ]
          },
          0
        ]
      }
    }
  }
])

The result will be:

  {
    "key": 1,
    "value": 10
  }

But if you change 20 to, say 44, the result is:

  {
    "key": 1,
    "value": 0
  }

Here is a Playground for you: https://mongoplayground.net/p/Y9yO6v9Oca8