I'm trying to get the audio from this as a ringtone for my phone. How do I do that [preferably without jailbreaking my phone]?

 

First do you ahve a copy of that tone? Preferably in an mp3 or similar format? If not, you can try out http://www.filsh.net to create an audio file only for it. It is in a language other than English but quite easy to nut out.

Next follow the instructions at this http://www.cnet.com.au/how-to-make-your-own-iphone-ringtones-339291124.htm?omnRef=NULL to change that audio clip into a ringtone format. You can probably skip steps 3 and 4 as the audio you want it very short already. Hope that helps.

Edit: Assuming you meant an iPhone because of the jailbreak comment.