You can do this by $unwinding the updates array, sorting the resulting docs by date, and then $grouping them back together on _id using the sorted order.
db.servers.aggregate(
{$unwind: '$service.apps.updates'},
{$sort: {'service.apps.updates.date': 1}},
{$group: {_id: '$_id', 'updates': {$push: '$service.apps.updates'}}},
{$project: {'service.apps.updates': '$updates'}})