I'd like to visit Tikal, Guatemala, but according to Rome2Rio, it's going to cost me at least $600 US to fly, which I find to be an exorbitant amount. What is the cheapest way to get from anywhere in Costa Rica to Tikal, Guatemala in a reasonable amount of time (e.g. buses)?
There are many buses of various levels in Central America. The cheapest are the former US and Canadian school buses affectionately termed "chicken buses" because many local people travel with anything including their produce and livestock.
Costa Rica is the one country in Central America which doesn't seem to have chicken buses. At least not on the major route along the Panamerican Highway. They may have them on provincial routes that I never travelled along. But since you've specified that you want the cheapest travel from anywhere in Costa Rica, just start with the chicken buses from the Nicaraguan border.
Chicken buses do not have set departure times. You just go to where they depart from and get on the next one. You may be able to buy tickets in advance from a spruiker or at the bus station / terminal if you're getting on or changing buses in a city.
In Central America most buses do not travel at night. This includes luxury buses as well as chicken buses. Since this route is pretty long you should expect to spend at least one night, and possibly a couple along the way.
Managua is not the safest city in the world but you may need to change buses at Granada, which is an interesting colonial city with cheap/fun backpacker hostels. Further, Tegucigalpa is safer than Managua, but maybe still less safe than San Jose. San Pedro Sula in Honduras not far from the Guatemala border is also popular with backpackers so would be a place to consider breaking your journey.
Last time I visited Tikal, the route within Guatemala was a lot worse than the route through Belize. Since I was in Chiapas, Mexico I took the Belize route. It's some years ago so perhaps the route within Guatemala has been upgraded since those days.
Google Maps says the driving time is a bit shy of sixteen hours. I would only use that as a very rough estimate. You'll have to add time for waiting for buses, possible breakdowns, getting to/from/between bus stations, getting to and from your accommodation at one or two cities along the way, etc.
Chicken buses are extremely cheap. The bus tickets for the trip will probably only add up to the equivalent of a handful of dollars. You will likely spend more money on food and sleeping and just getting to the Nicaraguan border from within Costa Rica.