Grading and defect: This hadith is "sahih" through the narration of 'Abdullah bin Abi Najih. As for the narration of 'Ubaydullah bin al-Walid (al-Wassafi) from 'Abdullah bin 'Ubayd bin 'Umayr, there is disagreement in it over the name of the person who performed twenty-five Hajjs on foot. Some of those who recorded this report mentioned "al-Husayn bin 'Ali" in place of (his brother) al-Hasan, and some narrated it from al-Wassafi, from Ibn 'Umayr, from Ibn 'Abbas, making it a saying of Ibn 'Abbas.
Verification: All of this disagreement goes back to "'Ubaydullah al-Wassafi", and he is "da'if" by consensus. What appears most likely is that the mention of "al-Husayn" is closer to correct, although there is disagreement over the number of his Hajjs, as will be explained below. And Allah knows best.

Reference / Source: Al-Fakihi narrated it in "Akhbar Makkah" (839) from Muhammad bin Ishaq al-Sini, from Ya'la bin 'Ubayd, with this chain.

Reference / Source: Ibn Sa'd narrated it in "al-Tabaqat" (6/ 410), and Abu al-Qasim al-Baghawi in "Mu'jam al-Sahabah" (407) from Ahmad bin Muhammad al-Qattan. Both of them (Ibn Sa'd and al-Qattan) narrate it from Ya'la bin 'Ubayd and mention "al-Husayn bin 'Ali" in it in place of (his brother) al-Hasan. And both of them cited it in the biography of al-Husayn (may Allah be pleased with him).

Reference / Source: Al-Bayhaqi narrated it in "al-Sunan al-Kubra" (4/ 331), and Ibn 'Asakir (13/ 242), by way of Zuhayr bin Mu'awiyah, from 'Ubaydullah bin al-Walid al-Wassafi, from 'Abdullah bin 'Ubayd bin 'Umayr, from Ibn 'Abbas. In it "al-Hasan bin 'Ali" is mentioned, and it is made a saying of Ibn 'Abbas.

Reference / Source: Ibn 'Asakir (14/ 180) narrated from Mus'ab bin 'Abdullah al-Zubayri, who said: Al-Husayn (may Allah be pleased with him) performed twenty-five Hajjs on foot. So he mentioned al-Husayn in place of his brother al-Hasan.

Reference / Sources: Ibn Sa'd (6/ 410), Ibn Hibban in "Rawdat al-'Uqala'" (p. 62) and Ibn 'Asakir (14/ 180) narrated by way of Muhammad bin 'Ali al-Baqir, who said: "Al-Husayn bin 'Ali performed ten Hajjs on foot while his riding camels were led alongside him." (Note: Ibn Sa'd, and Ibn 'Asakir through his route, did not mention the number of Hajjs in their narration.)
Decisive statement: Muhammad bin 'Ali al-Baqir knew the affairs of his own household better than anyone, since al-Husayn was his grandfather; so his statement on this matter is the "'umdah" (the main authority).