I am attempting to show that $A_1=\{1,2\} \times \mathbb{N}$ and $A_2=\mathbb{N} \times \{1,2\}$ do not have the same order type under the dictionary order. Now $(2,1)\in A_2$ has no immediate predecessor, but an infinite amount of predecessors. On the other hand, everything in $A_2$ has an immediate predecessor, and finitely many predecessors.
So intuitively I want to say they are different because of this fact, but I am relying on the assumption that order preserving bijections preserve immediate predecessors.
I did see the other post about the same problem, but the answers and explanations don't seem to be resolving the issue for me. I guess I am asking is this a valid reason for saying that there is no order-preserving bijection between the two sets?
Order preserving bijections need not respect immediate successor/predecesor, but order isomorphisms do.
I take “order preserving bijection” to mean a function $f\colon X\to Y$ between (po)sets such that $f$ is bijective, and $x\leq y$ implies $f(x)\leq f(y)$. Consider the case of $X=Y=\{a,b,c\}$, $f$ the identity, $X$ ordered by $\leq$ with $a<b$ and $c$ incomparable to $a$ and $b$; and $Y$ ordered by $\preceq$ with $a\preceq c\preceq b$.
This is different from an order isomorphism, which requires not just $x\leq y\implies f(x)\leq f(y)$, but requires $x\leq y \iff f(x)\leq f(y)$.
So, $x$ is the immediate predecessor of $y$ if $x<y$, and for all $z$, $x\leq z\leq y$ implies $x=z$ or $z=y$ (that is, if $y$ covers $x$, $x\prec y$).
Suppose $f\colon P\to Q$ is an order isomorphism. If $x\prec y$, and $w\in Q$ is such that $f(x)\leq w\leq f(y)$, then there exists $z\in P$ such that $f(z)=w$ (since $f$ is a bijection), hence $x\leq z\leq y$, hence $x=z$ or $z=y$, hence $f(x)=w$ or $w=f(y)$, showing that $f(x)\prec f(y)$.
So your are correct that “immediate predecessor” is respected by an order isomorphism.
However, your argument isn’t quite right, since $(1,1)$ in $\mathbb{N}\times\{1,2\}$ does not have an immediate predecessor.
However, your are on the correct track. How many elements have no immediate predecessor in $\{1,2\}\times\mathbb{N}$? And how many have no immediate predecessor in $\mathbb{N}\times\{1,2\}$?