Could it be the alternator or the starter?

Turn on the head lights if they seem bright the battery should be OK, ideally measure the battery voltage at rest should be 12.6 volts with the engine running if the alternator is charging correctly the battery voltage should be 14.6 volts approximately. The MR2 has replaceable starter contacts and plunger, when they fail you often just get a clicking sound as you try to engage the starter.

stop using the car right away and take it to a garage. this could be a sign of poor lubrication to the engine due to low quality engine oil or no engine oil. That ticking noise can very well be your valves misplacing themselves which will lead to your engine frying itself. check quality of engine oil by checking the dip stick.