Potential Coppersmith's attack on RSA
I'll try to answer your questions. First I shall write  Coppersmith's Theorem.

Theorem.  Let $0&lt;\varepsilon&lt;1/d$ and $F(x)$ be a monic polynomial of degree $d$ with at least one root $x_0$ in  ${\mathbb{Z}}_N$ and $|x_0|&lt;X =\lceil 0.5N^{1/d-\varepsilon}\rceil.$  Then, we can find $x_0$ in time $poly(d,1/\varepsilon,\ln{N}).$

First, notice that in your polynomial you know the factorization of the modulus. But, in Coppersmith method, you can find a solution $\mod{N},$ even if you don't know the factorization of $N$ (e.g. if $N$ is an RSA modulus).


should the coefficient of $^3$ be 1?


Yes, in the theorem of Coppersmith we deal with monic polynomials.


should the  of all the coefficients of $$ and $∗81$ be 1?


I did not understand what you mean. The gcd of the coefficients of the polynomial with $n?$ 


If we know the factorization of 81∗, can we benefit from it?


$81*P$ is your modulus. So, no. You can apply Coppersmith without knowing the factorization of the modulus.


What is the problem with using the Coppersmith method for 
with 1000 digits?


No problem, if the assumption of the Theorem are satisfied.