Will my road bike work with aerobars?
Probably. What's gets missed nearly every time this question comes up is that the ability of your road bike to work in a triathlon type position is inversely proportional to the size of your bike. The bigger you are, the trickier it becomes.
Road bikes have saddles further back relative to the pedals compared to tri bikes, and the bigger the size, the further back they go. In smaller sized bikes, simply sliding the saddle forward within it's built in adjustment range is often enough to mimic a triathlon bike, but larger bikes might need a more forward saddle position than even specialty seat posts allow. At 5'4" it's pretty easy, at 6'4" it can be harder or not completely possible.
In either case, the rider must learn to rotate the pelvis to mimic the geometric changes of the frame. HERE is a blog on the details of triathlon bike posture. In almost all cases, you will need a proper aerobar specific saddle to take full advantage of the bars. Figure $60-$200 for a saddle.
We are going for the "Deep Lean" on triathlon bikes.
So first you need to know how to rotate, along with a saddle that facilitates this. Next you need to position the aerobars at an appropriate distance both back and forth and up and down. A typical, well-fitting road bike is likely to need aerobars positioned closer and lower than would result from simply bolting them on. In any event, a stem will cost from $30-$100.
In my opinion, the best aerobars for turning a road bike into an effective triathlon bike are Profile Design's T+ series in alloy. That's right alloy, not carbon. The alloy version of these bars have what I call a "free floating" elbow pad, which afford a great deal of fore and aft adjustability, something often needed when going the road bike to tri bike route. These bars will cost you about $150.
You now have the knowledge, the correct saddle in the correct position, and the stem and aerobar combination for comfort and aerodynamics. If you want to put the icing on your fit, you can optimize for crank length as described HERE, (or you can take the bike for a ride).
All of these changes should flow from information derived from a real session on a fit bike. Simply bolting aerobars on a road bike is the #1 reason for a terrible bike fit in all of bicycledom.
So a bike fit at $250 plus a saddle at $150, stem at $50, aerobars at $150 and crankset at $300. That's $900 to properly turn a road bike into a triathlon machine!!! $1000 if you need a forward seat post. That's a lot of money, so what does it get you?
It gets you probably 70-90% of the gains that a real tri bike would net you. If a tri bike is worth 2-3mph over a road bike (typical), then a properly set up road bike is worth 1.75 - 2.75mph. Also, at least half of the items you purchase would go towards any future triathlon bike. Your crank and saddle definitely go with you, and maybe the aerobars as well. Done correctly, your bike fit will inform not only the current bike set up, but also future purchases for a number of years. The only throwaway items are maybe a stem or seatpost.
So yes, you should do this. Understand it is a little more involved than simply buying $150 aerobars and bolting them on, but done correctly, the speed benefits are huge, handling downsides are vastly overstated, and much of your expenditure is money you wont need to spend again when you do buy your first tri bike.
Get the bike fit. Everything flows from the bike fit.