What is the meaning of the Hadith, "By Allah, other than Whom there is no god, verily one of you may perform the deeds of the people of Paradise until there is but an arm's length between him and it, then the Book (preordainment) overtakes him..."? And is it understood from this that a person's righteous deeds throughout their life will not intercede for them if they do wrong at its end?

The gist of the matter is that what counts are the endings. So, whoever has a good ending is among the people of Paradise, and if it is evil, he is among the people of Hellfire. The righteous fear an evil ending and ask Allah for steadfastness. Endings are the inheritance of beginnings. A person may do good outwardly, while harboring a hidden evil within, or he may do evil outwardly, yet possess a hidden commendable trait through which Allah, by His mercy, redeems him. The transition from evil to good is common, while the transition from good to evil is rare.