Outlook/Hotmail has recently been updated and I'm not sure whether it still works with Live Mail (I think LiveMail is an unsupported program now). You should have no problems accessing it through a browser window. I found that the email client on my LG Android phone stopped working with the Outlook email service, so now I use the Outlook app for Android which works fine. You may need to do the same on your iPhone - find the Outlook app in the Itunes store. It is free. You will need to re-enter your account details into the Outlook app, even though you think that nothing has changed. Something has changed behind the scenes at Microsoft, hence the need to recreate the accounts on your phone.
I use Outlook 2010 on my PC for offline/local access to my mail boxes, and this works fine with the updated Outlook/Hotmail. I don't think I needed to re-enter the account details in Outlook but can't remember for certain. If necessary, remove the old accounts and re-establish them as new accounts. All of your old emails will download over a few hours.
