Мне удалось пойти в музей. = I managed to go to the museum. Мне довелось пойти в музей. = I had an opportunity to go the museum. "Удалось" means you made some effort to do it / achieve it. "Довелось" means you didn't really plan it but just used an opportunity.

* Мне удалось пойти в музей. - in spite of some difficulties you managed to go to a museum. * Мне довелось пойти в музей - You went to a museum, which you wanted to visit very much. There is a very small difference though.