Update an irule remotely?

yes, you can make an update via iControl. In fact, the iRule editor uses icontrol (SOAP interface) to make these changes, but you can use the REST interface as well by issuing a PUT. Here's a REST example I wrote in python a while back:

import requests, json

def modify_rule(bigip, name, code):
    payload = {}
    payload['apiAnonymous'] = code
    bigip.put('%s/ltm/rule/%s' % (b_url_base, name), data=json.dumps(payload))
    print 'Modifying rule...'

mycode = 'proc sequence {from to} {\n\tfor {set lst {}} {$from <=$to} {incr from} {\n\t\tlappend lst $from\n\t}\n\treturn $lst\n}'

b = requests.session()
b.auth = ('admin', 'admin')
b.verify = False
b.headers.update({'Content-Type' : 'application/json'})
b_url_base = 'https://172.16.44.15/mgmt/tm'

modify_rule(b, 'ruleProcLib', mycode)