Microsoft have updated the Outlook.com app for Android devices. The app gives you access to email from Outlook.com and has a new UI design that looks very much like the modern Windows design. It has new features like threaded conversation, flagging and filters for un-read email.
The app still doesn’t have email previews like the gmail and email apps have in Android (from the pull down menu) but it is certainly an improvement on the previous version. The app is free from the Google Play Store.