Convert date from milliseconds to ISODate object

Actually, it is possible, the trick is to add your milliseconds time to a zero-milliseconds Date() object using syntax similar to:

dt : {$add: [new Date(0), "$time"]}

I modified your aggregation from above to produce the result:

db.events.aggregate(
    {
        $project : {
            _id : "$_id",
            dt : {$add: [new Date(0), "$time"]}
        }
    },
    { 
        $project : {
            _id : "$_id",
            date : { 
                hour : {$hour : "$dt"} 
            }
        }
    }
);

The result is (with one entry of your sample data):

{
  "result": [
    {
      "_id": ObjectId("532828ac338ed9c33aa8eca7"),
      "date": {
        "hour": 11
      }
    }
  ],
  "ok": 1
}

I assume there's no way to do it. Because aggregation framework is written in native code. not making use of the V8 engine. Thus everything of JavaScript is not gonna work within the framework (And that's also why aggregation framework runs much faster).
Map/Reduce is a way to work this out, but aggregation framework definitely got much better performance.

About Map/Reduce performance, read this thread.

Another way to work it out would be get a "raw" result from aggregation framework, put it into an JSON array. Then do the conversion by running JavaScript. Sort of like:

var results = db.events.aggregate(...);
reasult.forEach(function(data) {
    data.date = new Date(data.dateInMillionSeconds);
    // date is now stored in the "date" property
}

To return a valid BSON date all you need is a little date "maths" using the $add operator. You need to add new Date(0) to the timestamp. The new Date(0) represents the number of milliseconds since the Unix epoch (Jan 1, 1970) and is a shorthand for new Date("1970-01-01").

db.events.aggregate([
    { "$match": { "time": { "$gte" : 1395136209804, "$lte" : 1395192902825 } } },
    { "$project": { 
        "hour": { "$hour": { "$add": [ new Date(0), "$time" ] } }, 
        "day": { "$dayOfMonth":  { "$add": [ new Date(0), "$time" ] } },
        "month": { "$month": { "$add": [ new Date(0), "$time" ] } },
        "year": { "$year":  { "$add": [ new Date(0), "$time" ] } } 
    }} 
])

Which yields:

{
    "_id" : ObjectId("532828ac338ed9c33aa8eca7"),
    "hour" : 11,
    "day" : 18,
    "month" : 3,
    "year" : 2014
}