There is some information missing to provide you with the best advice. But here are some ideas
First, Thanks to Sumeet, his question on the size of the original media is important. If you are increasing the size of your original media to fit your timeline, it will not look great.
In general, if your png file is of a lower size than the timeline, go into photo shop and make it the size the timeline, and use photo shops tools to get it to look as good as you can. Because that's all text/vector like images, the new 'super resolution' within photoshop would help you make it bigger with more clarity. Either way photoshop tools will do a much better job especially if you're starting with low res and want to increase the quality.
Premier has little issue downscaling. so if your PNG is larger than the timeline, it should look much better than the sample you provided.
Can you confirm your timeline settings, what I see there looks like the PNG was dropped into a interlaced timeline and then exported to a progressive timeline. But I would need confirmations of your timeline settings and export settings to be more confident.
I will say that premier is capabile of signficantly better output than that for that style of logo. You are on the right track of looking into the cause. Hoepfully your source PNG is good quality to start with. (if not try and make it better in photoshop before importing).
last thought, look at the image DPI settings also; Sometimes the DPI is pretty low, and that makes it look not great. I dont know what the idea DPI is for video, btu I tend to do everything in 300. If yours is 72 and a small image to begin with, it might be the cause.
After you have a chance to consider the above feedback, I look forward to hearing more details (or that you solved it).