Instagram has a very restricted input mechanism as it tries to remain "real" and thus requires users to post every single image/video manually. Prior to its acquisition by Facebook, Instagram did have an API (Application Program Interface), the communication methodology between two systems. This allows everyone to automatic posting (eg via IFTTT etc) without human intervention. That is no longer in place, so the short answer is NO, you cannot automate posts entirely.
One option - which we are testing today - is Later which was previously known as Latergramme. Later allows you to schedule a reminder to post - it stores the image and the description you put against it, and then alerts you at the time you scheduled. A few simple clicks later and you have loaded it into the Instagram app and posted it. The benefit is that you can schedule posts for when you are unavailable, or schedule between different members of a team so that someone else, perhaps in a different time zone to you, can post while you sleep, for example.
Later has a web interface, as as app for iOS or Android. It shows you a calendar of scheduled posts, and its free account allows 30 posts a month, 1 user and 2 social profile, with the number of posts resetting on 1st each month. $19 a month gives you 250 posts a month, 2 users and 2 social profiles - plus it comes down to $16 a month if you pre-pay a year. Other pricing options are available.
In due course we will also look at competitors Schedugram, Crowdfire, Onlypult and Hootsuite.