I don’t know if you have misunderstood the prince’s needs. Do you want to add a new field with a ISODate type value?
If so, $inc 并不是用来做这个的。$inc can only be used to increase or decrease numeric type values, for example:
Original data:

{ "_id" : ObjectId("537eaa530989f15b7f41cedf"), "i" : 1 }

Operation is as follows:

db.test.update({ i : 1 },{ $inc : { i : 2 }})

The result is:

{ "_id" : ObjectId("537eaa530989f15b7f41cedf"), "i" : 3 }

If you want to add a new field, you can use $push, as explained in the above example:

db.test.update({ i : 3 },{ $push : {'time' : new ISODate("2014-05-23")}})

The result is:

{ "_id" : ObjectId("537eaa530989f15b7f41cedf"), "i" : 3, "time" : [  ISODate("2014-05-23T00:00:00Z") ] }