All perfect praise be to Allah, The Lord of the worlds. I testify that there is none worthy of worship except Allah and that Muhammad is His slave and Messenger.
We have not come across any statement of the scholars that includes a precise text about the borders that the Dajjaal will reach when he wants to enter Madeenah. But he does not reach Mount Uhud, as the authentic narration states that he descends behind Uhud (i.e. before reaching Uhud) and the angels will drive him away.
Abu Hurayrah, may Allah be pleased with him, narrated that the Prophet said, “The Dajjaal will come from the East with the intention of attacking Madeenah until he gets down behind Uhud. Then the angels will turn his face towards Syria, and there he will perish.” [Muslim and Ahmad]
It is known that the borders of Madeenah from the north extend to Mount Thawr because of the hadeeth which states, “Madeenah is a Haram (Sanctuary), and its Sacred Precincts extend from Mount ’Ayr to Mount Thawr.” [Muslim]
Thawr, as stated by the scholars of Hadeeth is a mountain in Madeenah behind Mount Uhud, three miles away from Madeenah. This is evidence that the Dajjaal does not exceed the Sacred Precincts of Madeenah when he wants to enter it.
With regard to the hadeeth mentioned in the question, that the Dajjaal ascends Uhud, then the answer is in the two following points:
1- Not all the narrations of the hadeeth state that the Dajjaal ascends Uhud; this was only narrated in some of them, the narration by Al-Haakim in Al-Mustadrak and the narration by Ahmad. However, the narration reported by Al-Bukhaari in Al-Adab Al-Mufrad, by At-Tabaraani in Al-Mu’jam Al-Kabeer, and by Abu Daawood At-Tayaalisi states, “Mihjan (the narrator) said, ‘The Prophet once took me by hand, and we went together to the top of Uhud. He looked down on Madeenah and said, 'Woe to a town whose people will abandon it when it becomes very prosperous. Then the Dajjaal will come to it and find an angel at each of its gates, so he will not enter it.'’” This narration states that Mihjan was the one who ascended Mount Uhud along with the Prophet and there he informed him about the time when the Dajjaal will come out, but he did not tell him that the Dajjaal would ascend Uhud.
2- There is a difference of opinion about the authenticity of the hadeeth. Al-Haakim and Ath-Thahabi classified it as authentic, as you mentioned in the question, but other scholars classified it as Dha’eef (weak) because there is an interruption in the chain of narration between Abdullah ibn Shaqeeq and Mihjan ibn Al-Adra’, may Allah be pleased with him. It is for this reason that Al-Albaani did not decisively assert that it is authentic, as he said after reviewing Al-Haakim and Ath-Thahabi: “It [the hadeeth] is as they said, if it is free from interruption between ‘Abdullah ibn Shaqeeq and Mihjan, then Rajaa’ ibn Abi Rajaa’ Al-Baahili was introduced in the narration by Ahmad and Hanbal… ”
Shu'ayb Al-Arna'oot, when reviewing the Musnad, said, “Its chain of narrators is weak because of interruption. Abdullah ibn Shaqeeq did not listen to the narration of Mihjan ibn Al-Adra’, and between them there is Rajaa' ibn Abi Rajaa’, as stated in the following chain of narrators, and the rest of its narrators are trustworthy according to the criteria of the Sahih hadeeth.”
Rajaa’ ibn Abi Rajaa’ is unknown; Ath-Thahabi said about him, “It is only ‘Abdullah ibn Shaqeeq who narrated from him; his rank according to Al-Haakim is "acceptable"; meaning: if the same narration is narrated by other chain of narrators."
Allah knows best.