If you have time and travel light, then the cheapest option is to take the bus from downtown Panama to Tocumen Airport. The bus drops you off at the main road from where you walk for 5-10 mins to get to the airport. A taxi ride can cost you easily upwards of 20$ whereas the bus ride takes only 75 cents.
I used the app ‘moovit’ to find a bus from Casco Viejo to the Airport. I walked till the bus station next to the ‘5 de mayo’ metro station. I had to wait for 5 minutes to get the bus (S-480 or S-482) to Airport. This bus turned out to be an express bus which stops you just shy of the airport. The bus station is called Aeropuerto-R.
You can see the Airport terminal from the bus stop. There is a sheltered and well-marked path towards the terminal.
The bus was very convenient and pocket friendly means of getting to the Airport.