Translation:Finally he said he accepts that he is forty years old.
Sure! I am going to start from the beginning of the word though for people reading this in the future. :)
Take the verb root and add the participle suffix: ol + duk = olduk
Now you have to add a possessive suffix for it to really have any meaning. In this case, the subject of the embedded clause is an understood o (which would have the form of onun if you chose to write it, which is 3rd person singular). So: olduk + -sI = olduğu
This is the direct object of the verb "kabul etmek" however, which means that you need the accusative (you are accepting a certain specific thing after all). When you add any case to something ending with the 3rd person possessive suffix, you always get a buffer letter -n.
This means: olduğu + -n + *-(y)I = olduğunu
I agree that the placement of the sonunda does not make it clear what the exact translation should be. Placed at the beginning, it could be:
[Sonunda kırk yaşında olduğunu] [kabul ettiğini söyledi]. (...finally forty...)
[Sonunda] [kırk yaşında olduğunu kabul ettiğini] [söyledi]. (...finally said...)
If you needed it to be really clear, it seems like there are a few possible places for it, depending on what he is "finally" doing.
[Kırk yaşında sonunda olduğunu] [kabul ettiğini] [söyledi].
He said he accepts that he is finally forty years old.
(This place looks wrong, but I don't think I can explain why.)
[Kırk yaşında olduğunu] [sonunda kabul ettiğini] [söyledi].
He said he finally accepts that he is forty years old.
[Kırk yaşında olduğunu] [kabul ettiğini] [sonunda söyledi].
He finally said he accepts that he is forty years old.