It’s absolutely okay to go to Tibet alone, because Tibet is a very suitable place to make friends. Of course, if you want to go alone, that’s no problem either.
If you plan to make friends on the road, it is safer to go to outdoor forums to find travel companions before departure, such as Mofang, Youduoduo, etc. If you find one, communicate online to try to have a preliminary understanding before departure. Of course, you can also stay in a youth hostel. There are a lot of information about friends or carpooling here, making it easy to find a team. If you charter a car, you must go with someone else, otherwise it will be difficult for one person to bear the cost.
You only have one week, so you can’t go too far away from Lhasa. When entering Tibet for the first time, it is necessary to spend two days in Lhasa to adapt. By the way, you can visit the Potala Palace, Jokhang Temple, Drepung Monastery, and Sera Monastery. For attractions outside Lhasa, Namtso is still worth visiting. It is usually a one-day trip. If you want to see the sunrise and sunset, sign up for a two-day trip. The online tour price is only about 300.
It takes about four days to go to Linzhi and Shannan, which is not bad, and Linzhi is famous for its peach blossoms in early April. However, this loop usually requires a chartered car, which is relatively expensive. Mount Everest takes four days to go to Shigatse. This route is very high in altitude, so you have to travel according to your ability, and chartered cars and tickets are relatively expensive. Introductions to the attractions and tour prices for the above routes can be easily found online.
I don’t know what your purpose is for going to Tibet. If you just want to change your mood, it’s actually not bad to stay in Lhasa. There are many temples to visit, or you can just go to the Encounter Wall in front of the Jokhang Temple to bask in the sun. Many people stayed in Lhasa for several months.
When traveling to Tibet in April, just check the weather and bring enough warm clothes. It's windy on the plateau, so it's best to wear a jacket. If you go to Mount Everest or spend the night in Namtang, you must bring a down jacket. Lhasa has a complete range of supplies and daily necessities can be purchased.
In terms of budget, in addition to round-trip travel expenses, the key lies in the accommodation and food conditions you choose. Beds in multi-person rooms in Lhasa usually cost around forty per night. A standard room costs more than 200 yuan. Eating is not expensive, and you can eat well for fifty or sixty a day. Going out for fun also requires a chartered car and tickets. Basically, it is necessary to prepare about 3,000 in a week. But April is not the peak tourist season, so it might be cheaper. Good luck.