At beginning of July I'll be going to Makkah and Medina for the very first time in my life Insha Allah. And I'll be doing Umrah. I have read and heard that the very first sight/look of al-Ka'ba, your Dua' is Mustajab. Does this mean that only one Dua' can be made and answered at the first sight of Ka'ba or does it mean that all Dua's I make in this moment will be answered Insha Allah? I want to take the opportunity of course during this blessed moment to make as much Dua' as possible. If only one Dua' is answered then I thought of making my Dua' as follows: "Allahumma Ejalnee Mustajabat al-Dua'". Sorry if I'm confusing you and maybe this question is silly. Is it wrong to say like this? I hope you would help me sort my thoughts and reply to my question soon since my trip for Umrah is getting very close.

Praise be to Allah, the Lord of the World; and may His blessings and peace be upon our Prophet Muhammad and upon all his Family and Companions.

It is likable to supplicate Allah asking goodness of this world and the Hereafter at the time of looking at Ka'bah. One may also raise his hands for making Du'a. There is no sound Hadith proves any special Du'a at this place. So, one should ask Allah for goodness in this life and in the next life, the most important is forgiveness and safety in both lives. A Muslim should memorize the Prophetic supplications that contain all kinds of virtues to supplicate with them in such excellent places. For example, one may make this Du'a اللهم إني أسألك العفو والعافية ('O Allah, I ask you for pardon and well-being'). Imam Ibn Majah reported from Anas that a person came to the Prophet (Sallallahu Alaihi wa Sallam) and said, "O Messenger of Allah! Which Du'a is more excellent?' The Prophet (Sallallahu Alaihi wa Sallam) said: 'Ask your Lord for pardon and well-being in this life and the next'. He came to him another day and asked the same question. The Prophet (Sallallahu Alaihi wa Sallam) answered the same. He came to the Prophet (Sallallahu Alaihi wa Sallam) third day and asked him the same. The answer was the same and then the Prophet (Sallallahu Alaihi wa Sallam) said: "if you have been granted the well-being in this world and Hereafter as well, you will be successful" . [The Hadith is Hasan].
It is reported in Sahih Muslim that if someone would embrace Islam the Prophet (Sallallahu Alaihi wa Sallam) would teach him the prayer and then order him to supplicate with these words "اللهم اغفرلي وارحمني واهدني وعافني وارزقني" ('O Allah! Forgive me, have Mercy upon me, guide me, provide me well being and grant me sustenance" . [Sahih Muslim]
Know that there is no sound Hadith proving that supplications are responded at the time of sighting al-Ka'bah. The Hadith saying so are weak. Even though, such places are most likely to be where supplications are answered. So, one should supplicate Allah as much as possible and should insist on asking Allah, believing that He will respond and He is All-Hearer and Ever-Near.
Allah knows best.