Answer: The Shulchan Aruch (OC 314:1) writes that one may crack open a barrel on Shabbos to get to the food inside. While the Mishna Berura (314:25) allows opening a container to access the food inside, he writes (340:41; Biur Halacha 340:14) that one mustn’t open a sealed letter on Shabbos as the envelope becomes a usable kli, though one may ask a non-Jewish person to do so if absolutely necessary. R’ Moshe Feinstein (Igros Moshe OC 5:21:5) also forbids opening post nowadays, writing that with modern day communication, it’s difficult to say that regular post is so important.
While one may read a personal letter, especially if it contains Shabbos related material (such as a devar Torah), one must not read any business correspondence on Shabbos. Bills, etc. are therefore muktza, irrespective as to whether they arrived on Shabbos or before. Such post is muktza on Shabbos (Mishna Berura 307:56; Igros Moshe ibid.), though one may move it out of the way kilachar yad (such as with one’s foot).
The Kaf Hachaim (OC 586:140) quotes the Tashbetz who writes that one may crack open a barrel to get a shofar. Based on this, R’ Yehoshua Neuwirth (Shemiras Shabbos Kehilchasa 9:3; 15:80) writes that R’ Shlomo Zalman Auerbach applied this halacha equally to non-food items. Thus, one can remove clothes or a newspaper, etc. from their plastic providing one doesn’t intend on reusing the plastic afterwards (See Chazon Ish, Shabbos 61:2).
R’ Neuwirth (9:n18; 28:n15) suggests that the Mishna Berura would agree that one may open a non-food package if they disposed of the wrapping and it was necessary for Shabbos.
The Shulchan Aruch (307:14) writes that there is no issue in reading a letter that came from outside the techum, although if it was brought specially for them then it is preferable not to touch it. The Shulchan Aruch Harav (OC 515:14) explains that while one cannot normally benefit from something brought from outside the techum, it does not make the item muktze (See Aruch Hashulchan OC 307:5). Likewise, the Mishna Berura (307:55) explains that as one does not specifically expect the letter to come on Shabbos, there is no concern that one is asking the postman to deliver it (See Machazeh Eliyahu 37).
Thus, R’ Neuwirth (ibid. 31:23) writes that one may benefit from items that arrived in the post on Shabbos, providing they didn’t specifically ask for it to be delivered then.
In conclusion, while bills, etc. are muktze on Shabbos, one may open a package that arrived on Shabbos if necessary, providing they threw the packaging away.