Ok, please understand your post is not well informed by your own admission, so my replies are based on a best guess of standard architectures.
Question 1: Can I gain access to my VPN with Junos Pulse?
Answer 1: Yes, you can but not by using the web address you currently know. It seems your company has some kind of authentication software, think SiteMinder and or Imprivata. This single sign on leads to a realm on the IVE which based upon how long you have had it is probably a Network Connect realm. This will not work, as Network Connect and Junos Pulse are distinct and unique tunneling solutions. So what you need to do is get your IT people to stand up a new Realm, configure it to use Junos Pulse and authenticate against your native Active Directory infrastructure. Tell them for security reasons to create a OU with only a few names for authentication which should minimize any security concerns they could have. Total technical work time to do this is less then 60 minutes.
Question 2: Does the Network Admin need to do some work?
Answer 2: Yes he does but it is minimal. See above.
Question 3: What documentation does he need to reference, assuming he has questions?
Answer 3: I would suggest he look at the Junos Pulse Secure Access Service Administrator Guide. Everything you need can be found on this page. http://www.juniper.net/techpubs/en_US/sa7.1/inform
Question 4: Any idea what I should ask for or relay to IT?
Answer 4: See Answer 1. Summary: new realm using Junos Pulse, source Ip or DNS name, new Active Directory Object, name of favorite place to have lunch.
Question 5: Is there an alternate solution I should be looking at?
Answer 5: If you are using iPad's then no. If you are not commited to iPads and may have to suffer Droids then you may want to consider VMWare Virtual Desktops or some kind of thin client application delivery system. Still if I was your engineer I would say no, stay on the Juniper SA's and but iPads.
I do hope you find some of this useful, and good luck.