If you are really interested in pursuing a career in fundraising of a museum, better let that interest be known at very early stage. For example, you can start attending exhibitions and seminars etc from school days, and know about how the system works. This can also help you know that what type of museums you would like to work, as it will gradually develop a sense of liking in some types of historical heritage more than others.
Start volunteering yourself at your local museum, especially during the fundraising season and fundraising events. This can help you prepare a platform for launching yourself into the market properly. For example, it can help you build a contacts circle, who at some stage might be able to help you getting a job in the museum fundraising department. Also, your local museum will know you and they can let you at the time of arising of an opportunity.
Volunteering for donations collection on the street with wearing gear of the museum's name and details can help make people aware of you as an employee of the museum. Also, the museum's management will know you as a person who has an experience of fundraising and donations collection and they can offer you a permanent role in the department. Donating to your local museum can also market yourself as a person who has genuine interest in history and heritage.
Having proper qualification in the subject area can also help you land a job with the local museum fundraising department quickly. It is not necessary that you have the education and experience in the field of history and heritage, a good marketing person can also help the institution raise some good funds. You can be one of such potential people who are good at marketing of this important part of human ecology and get money out of people's pocket for the cause.
Whatever path you choose to fulfil your passion of working in a fundraising department of a museum, you should plan about the career at very early stage. It is not that the jobs in these institutions come up too often, and you might have to wait for some time, or else use your contacts or techniques that can help you quickly get into the role somehow. For this you should plan quite early and then take it step by step, until you become a regular employee at your local museum.