Ibn Abi Dunya recorded this Hadeeth, with this wording through a chain to Anas bin Malik, may Allaah be pleased with him, from the Prophet (sallAllaahu `alayhi wa sallam).
In the chain is Sulaiman bin Zaid Al-Ka'bi, who is weak in narrating. It was also recorded by Abu Dawud At-Tayalisi with a chain to 'Umar, may Allaah be pleased with him, but in its chain is someone who is unknown.
With this, there are authentic Hadeeths that stress the importance of visiting the graves in general, for reflecting upon death and the Hereafter. As for the Hadeeths mentioned about visiting the Prophet's grave in specific, then all of them are weak. They even say that they are fabricated.
If one wants to visit a grave, or to visit the Messenger (sallAllaahu `alayhi wa sallam), doing so according to the Shari'ah, reflecting upon death and the Hereafter, and supplicating for the dead, sending Salah upon the Prophet (sallAllaahu `alayhi wa sallam), and asking Allaah Almighty to be pleased with his two Companions, without taking up his mount or setting out to travel to do so - then his visit will be lawful and it is hoped that he will be rewarded.
Whoever takes up his mount to get to it, or sets out to travel to it, or visit it, hoping to gain blessings and benefit by it, or because he makes some type of routine visit to the grave specifically, then this visit is an innovation, there is no authentic text for it, and it is not known from any of the Salaf of this Nation.
Rather, there is proof to indicate the contrary. The Prophet (sallAllaahu `alayhi wa sallam) said,
لا تشد الرحال إلا إلى ثلاثة مساجد مسجدي هذا ومسجد الحرام ومسجد الأقصى
"Do not mount up except to three Masjids: Al-Masjid Al-Haram (in Makkah), this Masjid of mine (in Al-Madinah), and Al-Masjid Al-Aqsa (in Jerusalem)." (Al-Bukhari no. 1189 and Muslim no. 1397)
Yet another Hadeeth the Prophet (sallAllaahu `alayhi wa sallam) said,
لا تتخذوا قبري عيدا ولا تجعلوا بيوتكم قبورا وحيثما كنتم فصلوا علي فإن صلاتكم تبلغني
"Do not make my grave an 'Id (a place that you habitually visit), and do not make your homes into graves (by not praying and remembering Allaah in them). And send Salah upon me, for verily your greetings reach me, no matter where you are." (Abu Dawud no. 2042 and Ahmad 2/367)