I can read English written in the 1800s easily, but English in the 1600s is hard to understand. Would someone living in the 1800s find Shakespeare as hard to understand as I do?

I would first of all caution that it is difficult to generalise "Shakespeare" as a variety or chronolect of English, just because there is significant variety within his corpus. With some kinds of mannered speech (so called Euphuism) being intentionally obscure, while other speech is riddled with dialectal features and slang and neologisms, while other speech is fairly formal and devoid of uncommon (or outright invented) words.

So for example, when this (from Love's Labour's Lost) comes off as a pile of pretentious nonsense, that is because it is being spoken by a character who speaks pretentious nonsense and intentionally injects obscurities into his speech:

HOLOFERNES
The deer was, as you know, sanguis, in blood; ripe
as the pomewater, who now hangeth like a jewel in
the ear of caelo, the sky, the welkin, the heaven;
and anon falleth like a crab on the face of terra,
the soil, the land, the earth.

But more to the point and more generally, this writing is for the most part at least representative of the speech of various persons. Not of formal or narrative writing.

Note that if we take a formal translation of descriptive, narrative text from this same period (from the Authorized Version, i.e. KJV), the result is rather easily comprehensible to the modern reader (even if the subject is a bit obscure):

1 Samuel
1 Now there was a certain man of Ramathaimzophim, of mount Ephraim, and his name was Elkanah, the son of Jeroham, the son of Elihu, the son of Tohu, the son of Zuph, an Ephrathite:

2 And he had two wives; the name of the one was Hannah, and the name of the other Peninnah: and Peninnah had children, but Hannah had no children.

3 And this man went up out of his city yearly to worship and to sacrifice unto the LORD of hosts in Shiloh. And the two sons of Eli, Hophni and Phinehas, the priests of the LORD, were there.

4 And when the time was that Elkanah offered, he gave to Peninnah his wife, and to all her sons and her daughters, portions:

As to whether anything would be gained in the comprehension of Shakespeare by one's being a person living in the 1800s, I would say the largest factor determining comprehension is going to be the same one determining it today - education. And this is a greatly more important factor than one's year of birth within the period extending from 1800 to now.

There are individuals today who grow up with a very good education in or exposure to historic features of English and the uncommon (even then) words present in Shakespeare. And there were individuals of the early 19th century who grew up with essentially no exposure to those features and words at all.

But as I say, some of Shakespeare's writing was relatively opaque and mannered and obscure, or relatively informal and dialectal and riddled with slang, or what have you, even in his day.