Can I set a time limit for a push notification?

Yes, you can set an expiration time when publishing a push notification via APNS. Timeouts are measured in seconds (default is 3600, or 1 hour) and are set via the top level key of pn_ttl. For example, to set the timeout to 60 seconds (1 minute):

{
    "pn_apns": {
        "aps" : {
            "alert" : "Portugal vs. Denmark"
            "body":"Denmark - gooooooaaalllll!"
        }
    },
    "pn_gcm" : {
        "notification": {
            "title":"Portugal vs. Denmark",
            "body":"Denmark - gooooooaaalllll!"
        }
    },
    "pn_ttl": 60
}

Note that Apple will only queue a single message per device/app. For example, if you send 10 push notifications to a device that's powered off, then all notifications besides the last one are discarded, regardless of TTL.

(If a negative integer is entered for TTL value, the value is ignored and defaults to retention policy in Storage settings.)