Yes, the Surface Pro 3 will open any Microsoft Office docs from any version of Microsoft Office that originally came from Windows XP.
Any newer version of Microsoft Office will support any Office documents created by an older version of Microsoft Office. However, because of some changes over the years, a document formatted in Word 2003, for example, may not look the same in Word 2013. For instance, if you create a 2003 document that completely maximizes the page and fits on a single page, when you load that document in Word 2013, it may not fit properly.
If your real question is whether you can install your old Office 2003 on your Surface Pro 3, then that's a "yes" as well. However, a lot has changed since Office 2003. You might want to consider a newer version that makes some things a lot easier to do. Plus, Office 2013 is very touch friendly (when set to touch mode). I use Office 2013 extensively on my Surface Pro 2 with just touch and keyboard (no mouse involved). The same with my Surface RT. They did away with a lot of older habits so you can just tap, tap, tap to do things (no long holds which I absolutely hate and is a remnant of trying to emulate the right mouse click). Plus you can easily pinch to zoom in or out. And 2013 has much better pen support (though i believe 2003 had rudimentary support but don't hold me to that; it might have come with Office 2007).
If you don't want to spend the high price for a single version of Office that you have to hold on forever (like still using Office 2003 11 years later), you can always "lease" Office like Office 365. At my house, we have 5 PC installations of Office for $99/year. So with the subscription, you're always entitled to the latest version and upgrades to Office as they come out. You're not stuck to a single version indefinitely. So right now, I have the full suite of Office 2013. But when Office 2014 comes out or Office 2015 comes out, I have those versions as well.
Plus, you get all the online templates included so if you need a template for a cover letter or a fax letter, just browse them and get all of them.
But there's a personal option which is one installation for $69/year so that's cheaper. $5.75/month is not bad. I spend more at Starbucks a week than that. But the greater value is the $99/year because it's 5 machines. And those 5 machines don't have to be in the same house. If you have kids away at college, you can give each of them a license. Heck, if you want to share a subscription with friends, you can do that as well (though legally, I'm pretty sure you're not supposed to do that). At 5 license, that breaks down to $1.67/month per license.