Small riders need small bike with small wheels. MTB wheels are ideal, you can fit fast, slick road tyres and use it for fitness and commuting.
The best type of bike for your budget is probably a decent quality used non-suspension MTB from a respected brand (Trek, Specialized,Giant, GT etc), the kind that people rode in the 1990s.
The fit is important: MTBs are designed for more generous standover clearance than road bikes so look for a couple of inches of air between you and the top tube. More critical is the length of the reach from saddle to bars.
Budget for helmet, fenders, luggage rack, lights as required. With any used bike you may have to budget for replacement cogs,chain, tyres,cables if these are worn.