Arrive in San Jose late PM
Day in San Jose
Don't bother with a day in San Jose. Like most capital cities in Central America, it 's not chock full of things for tourists.
San Jose to Monteverde - Spend half a day
It's not adviseable. It will take you 4 hours to reach Monteverde from San Jose. Monteverde is reached via twisty slow mountain roads. Because of the transport time, if you go there, you'll probably want to stay 2 nights.
Monteverde to Arenal/ La Fortuna - Stay for 3 days
It will take you another 4 hours to reach La Fortuna starting from Monteverde.
1) would we need a car?
To go to Monteverde the public transport is extremely difficult, so yes you'll need a car. Also a car means you can go from La Fortuna to the la Pavona boat landing for Tortuguero much faster. If you used bus it would take 2x as long and you'd probably need to go through San Jose. And the car will need to be parked at the boat landing, since only boats go to Tortuguero. Car rental in CR isn't expensive if you reserve in advance, but you will need to pay for insurance at the rental car office, and the insurance costs more than the car rental.
3) is Tortuguero a beach area? if not is there anything close by instead of going to Puerto Viejo de Talamanca
There are beaches at Tortuguero. Sort of wilderness beaches with no services of any kind. Cahuita would be the nearest real beach town, way on the other side of Limon.