Outlook.com account doesn’t sync Drafts folder (this computer only)

You can't when an Outlook.com account is configured as an EAS account in Outlook 2013 as an EAS account doesn't support synching the Drafts folder.

There are a few workarounds for this available but depending on your needs and expectations, they might not all be viable solutions to you.

Update:
This workaround is no longer needed as Outlook.com accounts now connect as proper Exchange accounts (which supports synching the Drafts folder). For details see: Adding an Outlook.com account to Outlook.

Configure your Outlook.com account as an IMAP account

Instead of connecting your Outlook.com account via the EAS protocol, you could configure it as an IMAP account instead.

While it does support synching the Drafts folder, the IMAP protocol doesn't support a whole lot of other features such as Flag Reminders, Categories or synching Contacts, Calendar and Tasks Folders.

For the settings required to configure your Outlook.com mailbox as an IMAP account see:Configure Outlook with an Outlook.com (Hotmail), Gmail, Yahoo, AOL or iCloud account.

Send to yourself and Resend

Another workaround would be to send the message to yourself and then use the “Forward” or “Resend This Message” to continue writing in it and sending it to the correct recipient when you are done.

To resend a message in Outlook, double click it to open it in its own window and then from the Move group use Actions-> Resend This Message.

This is not possible in Outlook.com but you can use there and remove the “FW:” prefix from the subject line and the quoted header from the message itself.

To prevent cluttering your own Inbox with Draft messages, you can create a rule in Outlook.com to move messages which are From and To yourself to another folder.

To create a rule in Outlook.com, click on the gear icon in the top right corner (left from your picture and name) and choose “Manage rule” and press the “New” button.

While this messes a bit with the original way that Drafts should work, it should be quite workable unless you are a heavy Drafts user.

Setting up a rule in Outlook.com to move Draft sent to yourself. (click on image to enlarge)

Using OneDrive, Dropbox or OneNote

Yet another workaround would be to drag and drop the message out of Outlook and into your OneDrive or Dropbox folder. This will create a copy of the draft message in the msg-format which you can finish on another computer that also synchs with that OneDrive or Dropbox account and uses Outlook.

The downside of this method is that the message won’t be available in your Outlook.com mailbox online and that it only works with Outlook since the message is stored in the msg-format.

To counter the file format issue, you could compose your message in or copy its contents into a OneNote notebook. This notebook would is also available online and can sync with all popular smartphones and tablets. Once you are done, you can copy the contents from OneNote into a message and send it.

Extra tip: A quick way to get a Draft message into OneNote would be to select it in the Drafts folder and then use the OneNote button on the Home tab.