Actually the LTC4425 is a great solution to this type of problem and the chip is only 5 dollars from Digikey and the supporting components can be had for less than 2 dollars. Other solution could easily costs 5 dollars in supporting components. This chip will work with a single 5v capacitor with the addition of two small capacitors. Check out Linears web site and check ofout the datasheets. There are numerous other chips design for this purpose and some of them may also deal with your other problem which is routing power to the load from either the capacitor or the USB power source.
Most of my designs start out with looking for something that already exists that does all or as much possible of what I need it to do. I am currently work on project that does not use supercapacitors but instead using large conventional electrolytic capacitor to provide power to a load during startup in an automobile. I simply used the capacitor with nothing else before and it worked well but the inrush current is a problem.
My first design was focused on just dealing with the inrush current. My latest design takes a more comprehensive approach. In an automotive environment you also need to take into consideration the much higher than 12V can be supplied to your circuit at times and with high current levels available to fry things up crispy like. My new design also take into consideration protected the load and the capacitor for over-voltage events.
Take advantage of the work other very smart engineers have put-in to make your task easier. There is still plenty to learn from building something that someone else put the time and money into to figure out how it do it right. And it gives you time to thing of what else you might want to circuit to do.