Am I permitted to donate money on Shabbos to the shul etc?

First we must understand the nature of the prohibitions that are involved with donating money to shul on Shabbos and then see whether they can be overcome.

The first problem would seem to be the mentioning of money on Shabbos.

The Mechaber says [1] that it is permitted to discuss heavenly matters on Shabbos, which includes donating money to charity. Not only may one discuss on Shabbos whether certain poor people are in need of money and how much, one may even state the amount one intends to give them. All this is under the umbrella of çôöé ùîéí – heavenly matters, which one may deal with on Shabbos.

But is there not a problem of donating an item to a shul etc?

The Mechaber writes [2] that one may not dedicate an item to the Beis Hamikdash on Shabbos, [3] and the M”B explains that the issur involved is that it is similar to buying and selling. This is because dedicating ('makdish') an item to the Beis Hamikdash automatically activates the transaction and the dedicated item becomes the property of the Beis Hamikdash.

The same would apply to donating an item to a shul then, would it not?

Yes and no.  The Magen Avraham [4] says that (according to the Beis Yosef) it is prohibited to donate a specific item to a shul on Shabbos because it is similar to dedicating an item to the Beis Hamikdash.

He continues however that the Kol Bo says that one may donate an item to shul, and the Magan Avraham explains the difference between donating to the Beis Hamikdash and to shul saying that an item belonging to the Beis Hamikdash belongs only to the Beis Hamikdash and therefore dedicating an item to hekdesh (the property of the Beis Hamikdash) involves a complete transaction, whereas an item belonging to a shul belongs to the public as well and therefore when dedicating an item to a shul one has not accomplished a complete transaction being that the donor still owns a share in the item (together with the rest of Am Yisroel).

Which opinion is the halacha?

The Shulchan Aruch HaRav [5] writes that people do dedicate items to shul, but he concludes that one should dedicate it before Shabbos and bring it (where there is an eiruv) to shul on Shabbos or declare on Shabbos that it has been donated to the shul. The Mishna Berura reiterates this p’sak. [6]

This would not be a problem with money would it?

This problem does not exist with money or with non-specific items, i.e one may says on Shabbos that he will dedicate a paroches to the shul because one is not dedicating a specific ready-made article. The same applies with money. One may say that one intends donating 100$ to the shul because no transaction takes place and the mentioning of the money is not a problem, as stated, because it is for a tzedakah. [7]

Is one permitted to measure on Shabbos?

One is prohibited to take measurements on Shabbos, even though in essence it does not involve a melacha, because it is a weekday activity ('ovda d'chol') and is disrespectful to Shabbos, [8] and therefore it is not permitted on Shabbos.

What does this include?

This includes weighing items on a scale (mechanical obviously), weighing oneself on a scale, using a tape measure or ruler to measure distances, using a measuring spoon, measuring with a scaled glass or bottle. It is forbidden to measure the volume of a certain liquid, and therefore one may not pour liquid into a baby’s bottle and know its volume.

Rav Shlomo Zalman Auerbach ztz”l is in doubt whether counting floor tiles to know the size of a room is a problem, when one knows the width of an individual tile and thus know the size the room. [9]