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I'm looking for a reliable first car that is cheap to run, budget is <$5000.
I'm deciding between a 7th gen Toyota Corolla
http://en.wikipedia.org/wiki/Toyota_Corolla#Seventh_generation_.E2.80.94_E100_series_.E2.80.94_June_1991

or a 5th gen Honda Civic
http://en.wikipedia.org/wiki/Honda_Civic#Fifth_generation_.281992-1995.29

I like the Civic's body style better, but in terms of general
reliability, which one would you choose?  

well, since nobody has chimed in yet, I suppose there's no real difference in reliability between the two.  Both are highly reliable and long-lived, which is especially important in your case since you're probably looking at something with nearly 100K miles.  

The real difference is in the way they drive.  The corolla is the softer, less frisky of the two, though 7th generation Corollas, in my experience, were a bit less dull than 6th, 8th, and 9th gen.  

The Honda's definately the "performance" option, though don't expect anything truly sporty.  Then again, EX models come with the VTEC equipped 1.6 liter, which produces 125HP--modest by today's standards, but the engine is rev-happy and punchy in such a lightweight car.  

I forgot to mention that, since you're looking at high-mileage small sedans, take a look at the previous-generation Hyundai Elantra.  It seems to be highly reliable, and because it was cheaper to begin with, and doesn't hold its resale value as well as toyota and honda, you're likely to finder a newer, lower-mileage example that'll be similarly reliable.  That particular generation Elantra is also somewhat on par with the Corolla and Civic as far as refinement and fun-to-drive, in part because it's much newer than either.