I just upgraded someone to a 2-in-1 laptop with limited hard-drive space. We used a Google Email account, and had no issues with storage space. I'm trying to move one of my older in-laws away from WLM as well – it isn't supported very well by Microsoft – but they don't want to learn anything new.
If you want to keep using a Microsoft account, I would start up an "outlook.com" email account, and transfer everything to there. Microsoft has instructions here. It might be your easiest migration path.
http://windows.microsoft.com/en-us/outlook/import-desktop-app
From the link:
Import email from a desktop app
If you've been using a desktop app to manage your mail, like Outlook, Outlook Express, Windows Live Mail, or the version of Windows Mail that came with Windows Vista, your email messages and contacts are on the hard drive of your old PC. You can import them into Outlook.com using the Mail Migration add-on.
Import your email and contacts
On the PC that has your email and contacts, install the Mail Migration add-on for Internet Explorer.
Choose which accounts you want to import.
By default, the add-on will import all email and contacts from all email accounts on your PC.
Click Import.
Set up your Outlook.com account in the Mail app
If you liked using Outlook, Outlook Express, Windows Live Mail, or Windows Mail, and your PC runs Windows 8.1 or Windows RT 8.1, try the Mail app.
For more help with adding email accounts, contact customer support.
Welcome to Super User! Please quote the essential parts of the answer from the reference link(s), as the answer can become invalid if the linked page(s) change. – DavidPostill – 2015-09-13T08:02:14.077
You mean the Metro Mail App? That is a toy for me... I cannot add attachments by drag-and-drop, always fullscreen etc. – Louis Lam – 2015-09-13T09:13:56.990
This should be used in a browser such as Chrome/Firefox and possibly IE11 and MS Edge – depending on the browser, you can add attachments by drag-and-drop. For example, with Chrome/Gmail, I can copy an attachment from my desktop to my new mail message. Also, I tested this with Chrome/Outlook.com and it also worked. – RobEl – 2015-09-13T18:26:38.633