Rohirric, or rather Rohirian as Tolkien called it (the name Rohirric seems to be an invention of Robert Foster's from his 1970s Middle-earth guidebook) is not a language like Old English; rather it is akin to the Common Speech as well as the languages of Rhovanion, Esgaroth, and Dale. It is however represented in the books by the Mercian dialect of Anglo-Saxon (which of course we know the least about, harrumph!), just like the language of Dale is represented by Old Norse. As an example, I offer the name 'Hobbit' in English which relationship to the Old English "Holbytla" (Hole-builder) mirrors that of the Westron kuduk (the hobbit's own name for themselves) to the Rohirian kûd-dûkan (hole builder). Thus as no one would speak English in Middle-earth, the Rohirrim would not speak Mercian Old English but Rohirian. Excepting, of course, the good times when we converse in English as if in Middle-earth, then of course the Mercian names and sentences should be used as well.

As for the standard pronunciation of the Old English used to represent Rohirian, this is a good guide, and also explains the instances when Tolkien seems to have used pronunciations other than the conventional "scholarly" Anglo-Saxon one, but instead another for matter of aesthetics or personal lámatyávë (sound-taste).

And one more thing; if you want to write it like Tolkien did, transcribe the ð's and þ's you might encounter in Old English dictionaries or courses into th's. And also remember that sometimes the Mercian dialect differs from the typical West Saxon course you might take, for example where the common ðī­estru (darkness) is used but in Mercian it's ðēostru.